---—————-—--—-— —-«^^^ 

^ 

REESE  LIBRARY 

OF  THK 

UNIVERSITY  OF  CALIFORNIA.  -•; 
Glass 


APPLIED  MECHANICS. 


BY 


GAETANO    LANZA,    S.B.,  C.  &M.E., 

ii 

PROFESSOR  OF  THEORETICAL  AND  APPLIED  MECHANICS,  MASSACHUSETTS 
INSTITUTE    OF    TECHNOLOGY. 


NINTH  EDITION,  REVISED. 
FIRST    THOUSAND. 


UNIVERSITY 

OF 


NEW    YORK: 

JOHN   WILEY   &   SONS. 

LONDON  :  CHAPMAN  &  HALL,  LIMITED. 

1905. 


REESE 


c 

COPYRIGHT,  1885,  1900,  1905, 

BY 
GAETANO   LANZA. 


RObBKT   DRUMMOND,    1RIN1BK,    NBW   YORK. 


PREFACE. 


THIS  book  is  the  result  of  the  experience  of  the  writer 
in  teaching  the  subject  of  Applied  Mechanics  for  the  last 
twelve  years  at  the  Massachusetts  Institute  of  Technology. 

The  immediate  object  of  publishing  it  is,  to  enable  him  to 
dispense  with  giving  to  the  students  a  large  amount  of  notes. 
As,  however,  it  is  believed  that  it  may  be  found  useful  by 
others,  the  following  remarks  in  regard  to  its  general  plan 
are  submitted. 

The  work  is  essentially  a  treatise  on  strength  and  stabil- 
ity ;  but,  inasmuch  as  it  contains  some  other  matter,  it  was 
thought  best  to  call  it  "  Applied  Mechanics,"  notwithstanding 
the  fact  that  a  number  of  subjects  usually  included  in  trea- 
tises on  applied  mechanics  are  omitted. 

It  is  primarily  a  text-book ;  and  hence  the  writer  has  endeav- 
ored to  present  the  different  subjects  in  such  a  way  as 
seemed  to  him  best  for  the  progress  of  the  class,  even  though 
it  be  at  some  sacrifice  of  a  logical  order  of  topics.  While 
no  attempt  has  been  made  at  originality,  it  is  believed  that 
some  features  of  the  work  are  quite  different  from  all  pre- 


147GG3 


iv  PREFACE. 


vious  efforts ;  and  a  few  of  these  cases  will  be  referred  to, 
with  the  reasons  for  so  treating  them. 

In  the  discussion  upon  the  definition  of  "force,"  the  object 
is,  to  make  plain  to  the  student  the  modern  objections  to  the 
usual  ways  of  treating  the  subject,  so  that  he  may  have  a 
clear  conception  of  the  modern  aspect  of  the  question,  rather 
than  to  support  the  author's  definition,  as  he  is  fully  aware 
that  this,  as  well  as  all  others  that  have  been  given,  is  open 
to  objection. 

In  connection  with  the  treatment  of  statical  couples,  it 
was  thought  best  to  present  to  the  student  the  actual  effect 
of  the  action  of  forces  on  a  rigid  body,  and  not  to  delay  this 
subject  until  dynamics  of  rigid  bodies  is  treated,  as  is  usually 
done. 

In  the  common  theory  of  beams,  the  author  has  tried  to 
make  plain  the  assumptions  on  which  it  is  based.  A  little 
more  prominence  than  usual  has  also  been  given  to  the  longi- 
tudinal shearing  of  beams. 

In  that  part  of  the  book  that  relates  to  the  experimental 
results  on  strength  and  elasticity,  the  writer  has  endeavored 
to  give  the  most  reliable  results,  and  to  emphasize  the  fact, 
that,  to  obtain  constants  suitable  for  use  in  practice,  we 
must  deduce  them  from  tests  on  full-size  pieces.  This  prin- 
ciple of  being  careful  not  to  apply  experimental  results  to 
cases  very  different  from  those  experimented  upon,  has  long 
been  recognized  in  physics,  and  therefore  needs  no  justifica- 
tion. 

The  government  reports  of  tests  made  at  the  Watertown 
Arsenal  have  been  extensively  quoted  from,  as  it  is  believed 


PREFACE. 


that  'they  furnish  some  of   our  most  reliable  information  on 
these  subjects. 

The  treatment  of  the  strength  of  timber  will  be  found  to 
be  quite  different  from  what  is  usually  given ;  but  it  speaks 
for  itself,  and  will  not  be  commented  upon  here. 

In  the  chapter  on  the  "  Theory  of  Elasticity,"  a  combina- 
tion is  made  of  the  methods  of  Rankine  and  of  Grashof. 

In  preparing  the  work,  the  author  has  naturally  consulted 
the  greater  part  of  the  usual  literature  on  these  subjects ;  and, 
whenever  he  has  drawn  from  other  books,  he  has  endeavored  to 
acknowledge  it.  He  wishes  here  to  acknowledge  the  assist- 
ance furnished  him  by  Professor  C.  H.  Peabody  of  the  Massa- 
chusetts Institute  of  Technology,  who  has  read  all  the  proofs, 
and  has  aided  him  materially  in  other  ways  in  getting  out  the 
work. 

GAETANO  LANZA. 

MASSACHUSETTS  INSTITUTE  OF  TECHNOLOGY, 
April,  1885. 


PREFACE  TO  THE  FOURTH  EDITION. 

THE  principal  differences  between  this  and  the  earlier 
editions  consist  in  the  introduction  of  the  results  of  a  large 
amount  of  the  experimental  work  that  has  been  done  during 
the  last  five  years  upon  the  strength  of  materials. 

The  other  changes  that  have  been  made  in  the  book  are  not 
a  great  many,  and  have  been  suggested  as  desirable  by  the 
author's  experience  in  teaching. 

September,  1890. 


PREFACE   TO   THE   SEVENTH    EDITION. 


THE  principal  improvements  in  this  edition  consist  in  the 
introduction,  in  Chapter  VII,  of  the  results  of  a  considerable 
amount  of  the  experimental  work  on  the  strength  of  materials 
that  has  been  done  during  the  last  six  years.  A  few  changes 
have  also  been  made  in  other  parts  of  the  book. 

October,  1896. 


PREFACE 'TO   THE    EIGHTH    EDITION. 


IN  this  edition  a  considerable  number  of  additional  results 
of  recent  tests,  especially  upon  full-size  pieces,  have  been 
introduced,  some  of  the  older  ones  having  been  omitted  to 
make  room  for  them. 

September,  1900. 


PREFACE   TO    THE    NINTH    EDITION. 


THE  principal  improvements  in  the  Ninth  Edition  consist 
in  very  extensive  changes  in  Chapter  VII,  in  order  to  bring 
the  account  of  the  experimental  work  that  has  been  performed 
in  various  places  up  to  date. 

Some  changes  have  also  been  made  in  the  mathematical 
portion  of  the  book,  especially  in  the  Theory  of  Columns. 


TABLE  OF  CONTENTS. 


CHAPTER  I. 

COMPOSITION  AND  RESOLUTION  OF  FORCES  .    , 


CHAPTER  II. 
DYNAMICS .    •    .    •      75 

CHAPTER  III. 
ROOF-TRUSSES •«•»**,«••*•    138 

CHAPTER  IV. 
BBIDGE-TRUSSES 184 

CHAPTER  V. 
CENTRE  OF  GRAVITY 221 

CHAPTER  VI. 
STRENGTH  OF  MATERIALS 240 

CHAPTER  VII. 
STRENGTH  OF  MATERIALS  AS  DETERMINED  BY  EXPERIMENT     .....    350 


viii  TABLE   OF  CONTENTS. 

CHAPTER  VIII. 
CONTINUOUS  GIRDERS 743 

«r 
CHAPTER  IX. 

EQUILIBRIUM  CURVES.  —  ARCHES  AND  DOMES .   .    779 

CHAPTER  X. 
THEORY  OF  ELASTICITY,  AND  APPLICATIONS     ....    ••••«•<    852 


APPLIED    MECHANICS, 


CHAPTER   I. 
COMPOSITION  AND  RESOLUTION  OF  FORCES. 

§  i.  Fundamental  Conceptions.  —  The  fundamental  con- 
ceptions of  Mechanics  are  Force,  Matter,  Space,  Time,  and 
Motion. 

§  2.  Relativity  of  Motion.  —  The  limitations  of  our  natures 
are  such  that  all  our  quantitative  conceptions  are  relative. 
The  truth  of  this  statement  may  be  illustrated,  in  the  case  of 
motion,  by  the  fact,  that,  if  we  assume  the  shore  as  fixed  in 
position,  a  ship  sailing  on  the  ocean  is  in  motion,  and  a  ship 
moored  in  the  dock  is  at  rest ;  whereas,  if  we  assume  the  sun 
as  our  fixed  point,  both  ships  are  really  in  motion,  as  both  par- 
take of  the  motion  of  the  earth.  We  have,  moreover,  no  means 
of  determining  whether  any  given  point  is  absolutely  fixed  in 
position,  nor  whether  any  given  direction  is  an  absolutely  fixed 
direction.  Our  only  way  of  determining  direction  is  by  means 
of  two  points  assumed  as  fixed  ;  and  the  straight  line  joining 
them,  we  are  accustomed  to  assume  as  fixed  in  direction. 
Thus,  it  is  very  customary  to  assume  the  straight  line  joining 
the  sun  with  any  fixed  star  as  a  line  fixed  in  direction  ;  but  if 
the  whole  visible  universe  were  in  motion,  so  as  to  change  the 
absolute  direction  of  this  line,  we  should  have  no  means  of 
recognizing  it. 


APPLIED   MECHANICS. 


§3.  Rest  and  Motion.— -  In  order  to  define  rest  and 
motion,  we  have  the  following  ;  viz.,  — 

When  a  single  point  is  spoken  of  as  having  motion  or  rest, 
some  other  point  is  always  expressed  or  understood,  which  is 
for  the  time  being  considered  as  a  fixed  point,  and  some  direc- 
tion is  assumed  as  a  fixed  direction  :  and  we  then  say  that  the 
first-named  point  is  at  rest  relatively  to  the  fixed  point,  when 
the  straight  line  joining  it  with  the  fixed  point  changes  neither 
in  length,  nor  in  direction;  whereas  it  is  said  to  be  in  motion 
relatively  to  the  fixed  point,  when  this  straight  line  changes  in 
length,  in  direction,  or  in  both. 

If,  on  the  other  hand,  we  had  considered  the  first-named 
point  as  our  fixed  point,  the  same  conditions  would  determine 
whether  the  second  was  at  rest,  or  in  motion,  relatively  to  the 
first. 

A  body  is  said  to  be  at  rest  relatively  to  a  given  point  and 
to  a  given  direction,  when  all  its  points  are  at  rest  relatively  to 
this  point  and  this  direction. 

§  4.  Velocity  and  Acceleration. — When  the  motion  of 
one  point  relatively  to  another,  or  of  one  body  relatively  to 
another,  is  such  that  it  describes  equal  distances  in  equal  times, 
however  small  be  the  parts  into  which  the  time  is  divided, 
the  motion  is  said  to  be  uniform  and  the  velocity  constant. 

The  velocity,  in  this  case,  is  the  space  passed  over  in  a  unit 
of  time,  and  is  to  be  found  by  dividing  the  space  passed  over  in 
any  given  time  by  the  time ;  thus,  if  s  represent  the  space 
passed  over  in  time  /,  and  v  represent  the  velocity,  we  shall 
have 


When  the  motion  is  not  uniform,  if  we  divide  the  time  into 
small  parts,  and  then  divide  the  space  passed  over  in  one  of 
these  intervals  by  the  time,  and  then  pass  to  the  limit  as  these 
intervals  of  time  become  shorter,  we  shall  obtain  the  velocity 


FORCE. 


Thus,  if  A.y  represent  the  space  passed  over  in  the  interval  of 
time  A^,  then  we  shall  have 

v  =  limit  of  —  as  A/  diminishes, 
A/ 

or 

ds 


In  this  case  the  rate  of  change  of  velocity  per  unit  of  time 
is  called  the  Acceleration,  and  if  we  denote  it  by/,  we  have 


§  5.  Force.  —  We  shall  next  attempt  to  obtain  a  correct  defi- 
nition of  force,  or  at  least  of  what  is  called  force  in  mechanics. 

It  may  seem  strange  that  it  should  be  necessary  to  do  this  ; 
as  it  would  appear  that  clear  and  correct  definitions  must  have 
been  necessary  in  order  to  make  correct  deductions,  and  there- 
fore that  there  ought  to  be  no  dispute  whatever  over  the  mean- 
ing of  the  word  force.  Nevertheless,  it  is  a  fact  in  mechanics, 
as  well  as  in  all  those  sciences  which  attempt  to  deal  with  the 
facts  and  laws  of  nature,  that  correct  definitions  are  only  gradu- 
ally developed,  and  that,  starting  with  very  imperfect  and  often 
erroneous  views  of  natural  laws  and  phenomena,  it  is  only  after 
these  errors  have  been  ascertained  and  corrected  by  a  long 
range  of  observation  and  experiment,  and  an  increased  range  of 
knowledge  has  been  acquired,  that  exactness  and  perspicuity 
can  be  obtained  in  the  definitions. 

Now,  this  is  precisely  what  has  happened  in  the  case  of 
force. 

In  ancient  times  rest  was  supposed  to  be  the  natural  state 
of  bodies  ;  and  it  was  assumed  that,  in  order  to  make  them 
move,  force  was  necessary,  and  that  even  after  they  had  been 
set  in  motion  their  own  innate  inertia  or  sluggishness  would 
cause  them  to  come  to  rest  unless  they  were  constantly  urgea 


APPLIED    MECHANICS. 


on  by  the  application  of  some  force,  the  bodies  coming  to  rest 
whenever  the  force  ceased  acting. 

It  was  under  the  influence  of  these  vague  notions  that  such 
terms  arose  as  Force  of  Inertia,  Moment  of  Inertia,  Vis  Viva 
or  Living  Force,  etc. 

A  number  of  these  terms  are  still  used  in  mechanics;  but 
in  all  such  cases  they  have  been  re -defined,  such  new  mean- 
ings, having  been  attached  to  them  as  will  bring  them  into 
accord  with  the  more  advanced  ideas  of  the  present  time. 
Such  definitions  will  be  given  in  the  course  of  this  work,  as 
the  necessity  may  arise  for  the  use  of  the  terms. 

NEWTON'S    FIRST    LAW    OF    MOTION. 

Ideas  becoming  more  precise,  in  course  of  time  there  was 
framed  Newton's  first  law  of  motion  ;  and  this  law  is  as  fol- 
lows :  — 

A  body  at  rest  will  remain  at  rest,  and  a  body  in  motion  will 
continue  to  move  uniformly  and  in  a  straight  line,  unless  and 
until  some  external  force  acts  upon  it. 

The  assumed  truth  of  this  law  was  based  upon  the  observed 
facts  of  nature  ;  viz.,  — 

When  bodies  were  seen  to  be  at  rest,  and  from  rest  passed 
into  a  state  of  motion,  it  was  always  possible  to  assign  some 
cause ;  i.e.,  they  had  been  brought  into  some  new  relationship, 
either  with  the  earth,  or  with  some  other  body:  and  to  this 
cause  could  be  assigned  the  change  of  state  from  rest  to  motion. 
On  the  other  hand,  in  the  case  of  bodies  in  motion,  it  wa<>  seen, 
that,  if  a  body  altered  its  motion  from  a  uniform  rectilinear 
motion,  there  was  always  some  such  cause  that  could  be 
.assigned.  Thus,  in  the  case  of  a  ball  thrown  from  the  hand, 
the  attraction  of  the  earth  and  the  resistance  of  the  ait  soon 
caused  it  to  come  to  rest.  In  the  case  of  a  ball  rolled  along 
the  ground,  friction  (i.e.,  the  continual  contact  and  collision  with 
the  ground)  gradually  destroyed  its  motion,  and  brought  it  to 


FORCE. 


rest ;  whereas,  when  such  resistances  were  diminished  by  rolling 
it  on  glass  or  on  the  ice,  the  motion  always  continued  longer : 
hence  it  was  inferred,  that,  were  these  resistances  entirely 
removed,  the  motion  would  continue  forever. 

In  accordance  with  these  views,  the  definition  of  force 
usually  given  was  substantially  as  follows  :  — 

Force  is  that  which  causes,  or  tends  to  cause,  a  body  to  change 
its  state  from  rest  to  motion,  from  motion  to  rest,  or  to  change  its 
motion  as  to  direction  or  speed. 

Under  these  views,  uniform  rectilinear  motion  was  recog- 
nized as  being  just  as  much  a  condition  of  equilibrium,  or  of 
the  action  of  no  force  or  of  balanced  forces,  as  rest ;  and  the 
recognition  of  this  one  fact  upset  many  false  notions,  destroyed 
many  incorrect  conclusions,  and  first  rendered  possible  a  science 
of  mechanics.  Along  with  the  above-stated  definition  of  force 
is  ordinarily  given  the  following  proposition  ;  viz.,  — 

Forces  are  proportional  to  the  velocities  that  they  impart,  in  a 
unit  of  time  (i.e.  to  the  accelerations  that  they  impart),  to  the 
same  body.  The  reasoning  given  is  as  follows  : — 

Suppose  a  body  to  be  moving  uniformly  and  in  a  straight 
line,  and  suppose  a  force  to  act  upon  it  for  a  certain  length  of 
time  t  in  the  direction  of  the  body's  motion  :  the  effect  of  the 
force  is  to  alter  the  velocity  of  the  body ;  and  it  is  only  by  this 
alteration  of  velocity  that  we  recognize  the  action  of  the  force. 
Hence,  as  long  as  the  alteration  continues  at  the  same  rate,  we 
recognize  the  same  force  as  acting. 

If,  therefore,/  represent  the  amount  of  velocity  which  the 
force  would  impart  in  one  unit  of  time,  the  total  increase  in 
the  velocity  of  the  body  will  be  //;  and,  if  the  force  now  stop 
acting,  the  body  will  again  move  uniformly  and  in  the  same 
direction,  but  with  a  velocity  greater  by//. 

Hence,  if  we  are  to  measure  forces  by  their  effects,  it  will 
follow  that  — 

The  velocity  which  a  force  will  impart  to  a  given  (or  standard) 


APPLIED  MECHANICS. 


body  in  a  unit  of  time  is  a  proper  measure  of  the  force.  And 
we  shall  have,  that  two  forces,  each  of  which  will  impart  the 
same  velocity  to  the  same  body  in  a  unit  of  time,  are  equal  to 
each  other  ;  and  a  force  which  will  impart  to  a  given  body  twice 
the  velocity  per  unit  of  time  that  another  force  will  impart  to 
the  same  body,  is  itself  twice  as  great,  or,  in  other  words,  — 

Forces  are  proportional  to  the  velocities  that  they  impart,  in  a 
unit  of  time  (i.e.  to  the  accelerations  that  they  impart),  to  the 
same  body. 

MODERN   CRITICISM   OF   THE   ABOVE. 

The  scientists  and  the  metaphysicians  of  the  present  time 
are  recognizing  two  other  facts  not  hitherto  recognized,  and  the 
result  is  a  criticism  adverse  to  the  above-stated  definition  of 
force.  Other  definitions  have,  in  consequence,  been  proposed  ; 
but  none  are  free  from  objection  on  logical  grounds,  and  at  the 
same  time  capable  of  use  in  mechanics  in  a  quantitative  way. 

The  two  facts  referred  to  are  the  following ;  viz.,  — 

i°.  That  all  our  ideas  of  space,  time,  rest,  motion,  and  even 
of  direction,  are  relative. 

2°.  That,  because  two  effects  are  identical,  it  does  not  follow 
that  the  causes  producing  those  effects  are  identical. 

Hence,  in  the  light  of  these  two  facts,  it  is  plain,  that,  inas- 
much as  we  can  only  recognize  motion  as  relative,  we  can  only 
recognize  force  as  acting  when  at  least  two  bodies  are  con- 
cerned in  the  transaction  ;  and  also  that  if  the  forces  are  simply 
the  causes  of  the  motion  in  the  ordinary  popular  sense  of  the 
word  cause,  we  cannot  assume,  that,  when  the  effects  are  equal, 
the  causes  are  in  every  way  identical,  although  we  have,  of 
course,  a  perfect  right  to  say  that  they  are  identical  so  far  as 
the  production  of  motion  is  concerned. 

I  shall  now  proceed,  in  the  light  of  the  above,  to  deduce  a 
definition  of  force,  which,  although  not  free  from  objection, 
seems  as  free  as  any  that  has  been  framed. 

It  is  one  of  the  facts  of  nature,  that,  when  bodies  are  by  any 


FORCE. 


means  brought  under  certain  relations  to  each  other,  certain 
tendencies  are  developed,  which,  if  not  interfered  with,  will 
exhibit  themselves  in  the  occurrence  of  certain  definite  phe- 
nomena. What  these  phenomena  are,  depends  upon  the  nature 
of  the  bodies  concerned,  and  on  the  relationships  into  which 
they  are  brought. 

As  an  illustration,  we  know  that  if  an  apple  is  placed  at  a 
certain  height  above  the  surface  of  the  earth,  there  is  developed 
between  the  two  bodies  a  tendency  to  approach  each  other ; 
and  if  there  is  no  interference  with  this  tendency,  it  exhibits 
itself  in  the  fall  of  the  apple.  If,  on  the  other  hand,  the  apple 
were  hung  on  the  hook  of  a  spring  balance  in  the  same  posi- 
tion as  before,  the  spring  would  stretch,  and  there  would  be 
developed  a  tendency  of  the  spring  to  make  the  apple  move 
upwards.  This  tendency  to  make  the  apple  move  upwards 
would  be  just  equal  to  the  tendency  of  the  earth  and  apple  to 
approach  each  other.  This  would  be  expressed  by  saying  that 
the  pull  of  the  spring  is  just  equal  and  opposite  to  the  weight 
of  the  apple. 

As  other  illustrations  of  these  tendencies  developed  in 
bodies  when  placed  in  certain  relations  to  each  other,  we  have 
the  following  cases  :  — 

(a}  When  two  bodies  collide. 

(b}  When  two  substances,  coming  together,  form  a  chemical 
union,  as  sodium  and  water. 

(c)  When  the  chemical  union  is  entered  into  only  by  raising 
the  temperature  to  some  special  point. 

Any  of  these  tendencies  that  are  developed  by  bringing 
about  any  of  these  special  relationships  between  bodies  might 
properly  be  called  a  force  ;  and  the  term  might  properly  be,  and 
is,  used  in  the  same  sense  in  the  mental  and  moral  world,  as 
well  as  in  the  physical.  In  mechanics,  however,  we  have  to 
deal  only  with  the  relative  motion  of  bodies  ;  and  hence  we 
give  the  name  force  only  to  tendencies  to  change  the  relative 


8  APPLIED   MECHANICS. 

motion  of  the  bodies  concerned ;  and  this,  whether  these  ten- 
dencies are  unresisted,  and  exhibit  themselves  in  the  actual 
occurrence  of  a  change  of  motion,  or  whether  they  are  resisted 
by  equal  and  opposite  tendencies,  and  exhibit  themselves  in 
the  production  of  a  tensile,  compressive,  or  other  stress  in  the 
bodies  concerned,  instead  of  motion. 

DEFINITION    OF    FORCE. 

Hence  our  definition  of  force,  as  far  as  mechanics  has  to 
deal  with  it  or  is  capable  of  dealing  with  it,  is  as  follows; 
viz.,  - 

Force  is  a  tendency  to  change  the  relative  motion  of  the  two 
bodies  between  which  that  tendency  exists. 

Indeed,  when,  as  in  the  illustration  given  a  short  time  ago, 
the  apple  is  hung  on  the  hook  of  a  spring  balance,  there  still 
exists  a  tendency  of  the  apple  and  the  earth  to  approach  each 
other  ;  i.e.,  they  are  in  the  act  of  trying  to  approach  each  other  ; 
and  it  is  this  tendency,  or  act  of  trying,  that  we  call  the  force  of 
gravitation.  In  the  case  cited,  this  tendency  is  balanced  by 
an  opposite  tendency  on  the  part  of  the  spring ;  but,  were  the 
spring  not  there,  the  force  of  gravitation  would  cause  the  apple 
to  fall. 

Professor  Rarikine  calls  force  "an  action  between  two  bodies, 
either  causing  or  tending  to  cause  change  in  their  relative  rest 
or  motion  ;"  and  if  the  act  of  trying  can  be  called  an  action,  my 
definition  is  equivalent  to  his. 

For  the  benefit  of  any  one  who  wishes  to  follow  out  the 
discussions  that  have  lately  taken  place,  I  will  enumerate  the 
following  articles  that  have  been  written  on  the  subject :  — 

(a)  "  Recent  Advances  in  Physical  Science,"  by  P.  G.  Tait, 
Lecture  XIV. 

(b)  Herbert    Spencer,    "First    Principles    of    Philosophy* 
(certain  portions  of  the  book). 


MEASURE   OF  FORCE. 


(£)  Discussion  by  Messrs.  Spencer  and  Tait,  "  Nature,"  Jan. 
2,  9,  1 6,  1879. 

(d]  Force  and  Energy,  "Nature,"  Nov.  25,  Dec.  2,  9,  16, 
1880. 

§6.  External  Force.  —  We  thus  see,  that,  in  order  that  a 
force  may  be  developed,  there  must  be  two  bodies  concerned 
in  the  transaction ;  and  we  should  speak  of  the  force  as  that 
developed  or  existing  between  the  two  bodies. 

But  we  may  confine  our  attention  wholly  to  the  motion  or 
condition  of  one  of  these  two  bodies ;  and  we  may  refer  its 
motion  either  to  the  other  body  as  a  fixed  point,  or  to  some 
body  different  from  either;  and  then,' in  speaking  of  the  force, 
we  should  speak  of  it  as  the  force  acting  on  the  body  under 
consideration,  and  call  it  an  external  force.  It  is  the  tendency 
of  the  other  body  to  change  the  motion  of  the  body  under  con- 
sideration relatively  to  the  point  considered  as  fixed. 

§  7.  Relativity  of  Force.  —  In  adopting  the  above-stated 
definition  of  force,  we  acknowledge  our  incapacity  to  deal  with 
it  as  an  absolute  quantity ;  for  we  have  defined  it  as  a  tendency 
to  change  the  relative  motion  of  a  pair  of  bodies.  Hence  it  is 
only  through  relative  motion  that  we  recognize  force;  and  hence 
force  is  relative,  as  well  as  motion. 

§8.  Newton's  First  Law  of  Motion.  —  In  the  light  of 
the  above  discussion,  we  might  express  Newton's  first  law  of 
motion  as  follows  :  — 

A  body  at  rest,  or  in  uniform  rectilinear  motion  relatively  to 
a  given  point  assumed  as  fixed,  will  continue  at  rest,  or  in  uni- 
form motion  in  the  same  direction,  unless  and  until  some  external 
force  acts  either  on  the  body  in  question,  or  on  the  fixed  point, 
or  on  the  body  which  furnishes  us  our  fixed  direction.  This  law 
is  really  superfluous,  as  it  has  all  been  embodied  in  the  defini- 
tion. 

§9.  Measure  of  Force.  —  We  next  need  some  means  of 
comparing  forces  with  each  other  in  magnitude ;  and,  subse- 


10  APPLIED  MECHANICS. 

quently,  we  need  to  select  one  force  as  our  unit  force,  by  means 
of  which  to  estimate  the  magnitude  of  other  forces. 

Let  us  suppose  a  body  moving  uniformly  and  in  a  straight 
line,  relatively  to  some  fixed  point ;  as  long  as  this  motion 
continues,  we  recognize  no  unbalanced  force  acting  on  it ; 
but,  if  the  motion  changes,  there  must  be  a  tendency  to  change 
that  motion,  or,  in  other  words,  an  unbalanced  force  is  acting 
on  the  body  from  the  instant  when  it  begins  to  change  its 
motion. 

Suppose  a  body  to  be  moving  uniformly,  and  a  force  to  be 
applied  to  it,  and  to  act  for  a  length  of  time  /,  and  to  be  so  applied 
as  not  to  change  the  direction  of  motion  of  the  body,  but  to 
increase  its  velocity;  the  result  will  be,  that  the  velocity  will  be 
increased  by  equal  amounts  in  equal  times,  and  if  f  represent 
the  amount  of  velocity  the  force  would  impart  in  one  unit  of 
time,  the  total  increase  in  velocity  will  'be//.  This  results 
merely  from  the  definition  of  a  force ;  for  if  the  velocity  pro- 
duced in  one  (a  standard)  body  by  a  given  force  is  twice  as 
great  as  that  produced  by  another  given  force,  then  is  the  ten- 
dency to  produce  velocity  twice  as  great  in  the  first  case  as  in 
the  second,  or,  in  other  words,  the  first  force  is  twice  as  great 
as  the  second.  Hence  — 

Forces  are  proportional  to  the  velocities  which  they  will  impart 
to  a  given  (or  standard}  body  in  a  unit  of  time. 

We  may  thus,  by  using  one  standard  body,  determine  a 
set  of  equal  forces,  and  also  the  proportion  between  different 
forces. 

§  10.  Measure  of  Mass.  —  After  having  determined,  as 
shown,  a  set  of  equal  (unit)  forces,  if  we  apply  two  of  them 
to  different  bodies,  and  let  them  act  for  the  same  length  of  time 
on  each,  and  find  that  the  resulting  velocities  are  unequal,  these 
bodies  are  said  to  have  unequal  masses ;  whereas,  if  the  result- 
ing velocities  are  equal,  they  are  said  to  have  equal  masses. 

Hence  we  have  the  following  definitions  :  — 


RELATION  BETWEEN  FORCE   AND   MOMENTUM.  II 

I  °.  Equal  forces  are  those  which,  by  acting1  for  equal  times 
on  tJie  same  or  standard  body,  impart  to  it  equal  velocities. 

2°.  Equal  masses  are  those  masses  to  which  equal  forces 
will  impart  equal  velocities  in  equal  times. 

§11.  Suppose  two  bodies  of  equal  mass  moving  side  by 
side  with  the  same  velocity,  and  uniformly,  let  us  apply  to 
one  of  them  a  force  F  in  the  direction  of  the  body's  motion : 
the  effect  of  this  force  is  to  increase  the  velocity  with  which  the 
body  moves ;  and  if  we  wish,  at  the  same  time,  to  increase 
the  velocity  of  the  other,  so  that  they  will  continue  to  move 
side  by  side,  it  will  be  necessary  to  apply  an  equal  force  to  that 
also. 

We  are  thus  employing  a  force  2F  to  impart  to  the  two 
bodies  the  required  increment  of  velocity. 

If  we  unite  them  into  one,  it  still  requires  a  force  2F  to 
impart  to  the  one  body  resulting  from  their  union  the  re- 
quired increment  of  velocity :  hence,  if  we  double  the  mass 
to  which  we  wish  to  impart  a  certain  velocity,  we  must  double 
the  force,  or,  in  other  words,  employ  a  force  which  would 
impart  to  the  first  mass  alone  a  velocity  double  that  required. 
Hence  — 

Forces  are  proportional  to  the  masses  to  which  they  will  impart 
the  same  velocity  in  the  same  time. 

§  12.  Momentum.  —  The  product  obtained  by  multiplying 
the  number  of  units  of  mass  in  a  body  by  its  velocity  is  called 
the  momentum  of  the  body. 

§13.  Relation  between  Force  and  Momentum. — The 
number  of  units  of  momentum  imparted  to  a  body  in  a  unit  of 
time  by  a  given  force,  is  evidently  identical  with  the  number 
of  units  of  velocity  that  would  be  imparted  by  the  same  force, 
in  the  same  time,  to  a  unit  mass.  Hence  — 

Forces  are  proportional  to  the  momenta  (or  velocities  per  unit 
of  mass)  which  they  will  generate  in  a  unit  of  time. 


12  APPLIED  MECHANICS. 

Hence,  if  F  represent  a  force  which  generates,  in  a  unit  of 
time,  a  velocity/"  in  a  body  whose  mass  is  m,  we  shall  have 


and,  inasmuch  as  the  choice  of  our  units  is  still  under  our  con- 
trol, we  so  choose  them  that 

F  =  mf; 

i.e.,  the  force  F  contains  as  many  units  of  force  as  mf  contains 
units  of  momentum  ;  in  other  words,  — 

The  momentum  generated  in  a  body  in  a  unit  of  time  by  a. 
force  acting  in  the  direction  of  the  body's  motion,  is  taken  as 
a  measure  of  the  force. 

§14.  Statical  Measure  of  Force.  —  When  the  forces  are 
prevented  from  producing  motion  by  being  resisted  by  equal 
and  opposite  fofces,  as  is  the  case  in  that  part  of  mechanics 
known  as  Statics,  they  must  be  measured  by  a  direct  comparison 
with  other  forces.  An  illustration  of  this  has  already  been 
given  in  the  case  of  an  apple  hung  on  the  hook  of  a  spring 
balance.  In  that  case  the  pull  of  the  spring  is  equal  in  magni- 
tude to  the  weight  of  the  apple  :  indeed,  it  is  very  customary 
to  adopt  for  forces  what  is  known  as  the  gravity  measure,  in 
which  case  we  take  as  our  unit  the  gravitation,  or  tendemy  to 
fall,  of  a  given  piece  of  metal,  at  a  given  place  on  the  surface 
of  the  earth  ;  in  other,  words,  its  weight  at  a  given  place. 

The  gravity  unit  may  thus  be  the  kilogram,  the  pound,  or 
the  ounce,  etc. 

It  is  evident,  moreover,  from  our  definition  of  force,  and  the 
subsequent  discussion,  that  whatever  we  take  as  our  unit  of 
mass,  the  statical  measure  of  a  force  is  proportional  to  its 
dynamical  measure  ;  i.e.,  the  numbers  representing  the  magni- 
tudes of  any  two  forces,  in  pounds,  are  proportional  to  the 
momenta  they  will  impart  to  any  body  in  a  unit  of  time. 

§15.  Gravity  Measure  of  Mass.  —  If  we  assume  one 
pound  as  our  unit  of  force,  one  foot  as  our  unit  of  length,  and 


NEWTON'S  SECOND   LAW  OF  MOTION  13 

one  second  as  our  unit  of  time,  the  ratio  between  the  number 
of  pounds  in  any  given  force  and  the  momentum  it  will  impart 
to  a  body  on  which  it  acts  unresisted  for  a  unit  of  time,  will 
depend  on  our  unit  of  mass  ;  and,  as  we  are  still  at  liberty  to  fix 
this  as  we  please,  it  will  be  most  convenient  so  to  choose  it 
that  the  above-stated  ratio  shall  be  unity,  so  that  there  shall  be 
no  difference  in  the  measure  of  a  force,  whether  it  is  measured 
statically  or  dynamically.  Now,  it  is  known  that  a  body  falling 
freely  under  the  action  of  its  own  weight  acquires,  every  second, 
a  velocity  of  about  thirty-two  feet  per  second  :  this  number  is 
denoted  by  gt  and  varies  for  different  distances  from  the  centre 
of  the  earth,  as  does  also  the  weight  of  the  body. 

Now,  if  W  represent  the  weight  of  the  body  in  pounds,  and 
m  the  number  of  units  of  mass  in  its  mass,  we  must  have,  in 
order  that  the  statical  and  dynamical  measures  may  be  equal, 

W =  mg. 
Hence 

m.y, 

g 

i.e.,  the  number  of  units  of  mass  in  a  body  is  obtained  by  divid- 
ing the  weight  in  pounds,  by  the  value  of  g  at  the  place  where 
the  weight  is  determined. 

The  values  of  W  and  of  g  vary  for  different  positions,  but 
the  value  of  m  remains  always  the  same  for  the  same  body. 

UNIT    OF    MASS. 

If  m  =  I,  then  W  —  g;  or,  in  words,  — 

The  weight  in  pounds  of  the  unit  of  mass  (when  the  gravity 
measure  is  used}  is  equal  to  the  value  of  g  in  feet  per  second  for 
the  same  place. 

§  16.  Newton's  Second  Law  of  Motion.  —  Newton's 
second  law  of  motion  is  as  follows  :  — 


14  APPLIED  MECHANICS. 

"  Change  of  momentum  is  proportional  to  the  impressed  mov- 
ing f°rce>  and  occurs  along  the  straight  line  in  which  the  force  is 
impressed" 

Newton  states  further  in  his  "  Principia  :"  — 

"  If  any  force  generate  any  momentum,  a  double  force 
will  generate  a  double,  a  triple  force  will  generate  a  triple, 
momentum,  whether  simultaneously  and  suddenly,  or  gradually 
and  successively  impressed.  And  if  the  body  was  moving 
before,  this  momentum,  if  in  the  same  direction  as  the  motion, 
is  added;  if  opposite,  is  subtracted;  or  if  in  an  oblique  direc- 
tion, is  annexed  obliquely,  and  compounded  with  it,  according 
to  the  direction  and  magnitude  of  the  two." 

Part  of  this  law  has  reference  to  the  proportionality  between 
the  force  and  the  momentum  imparted  to  the  body ;  and  this 
has  been  already  embodied  in  our  definition  of  force,  and  illus- 
trated in  the  discussion  on  the  measure  of  forces. 

The  other  part  is  properly  a  law  of  motion,  and  may  be 
expressed  as  follows  :  — 

If  a  body  have  two  or  more  velocities  imparted  to  it  simulta- 
neously, it  will  move  so  as  to  preserve  them  all. 

The  proof  of  this  law  depends  merely  upon  a  proper  con- 
ception of  motion.  To  illustrate  this  law  when  two  velocities 
are  imparted  simultaneously  to  a  body,  let  us  suppose  a  man 
walking  on  the  deck  of  a  moving  ship  :  he  then  has  two  motions 
in  relation  to  the  shore,  his  own  and  that  of  the  ship. 

Suppose  him  to  walk  in  the  direction  of  motion  of  the 
ship  at  the  rate  of  10  feet  per  second,  while  the  ship  moves  at 
25  feet  per  second  relatively  to  the  shore  :  then  his  motion  in 
relation  to  the  shore  will  be  25  -|-  10  =  35  feet  per  second. 
If,  on  the  other  hand,  he  is  walking  in  the  opposite  direction  at 
the  same  rate,  his  motion  relatively  to  the  shore  will  be  25  — 
10  —  15  feet  per  second. 

Suppose  a  body  situated  at  A  (Fig.  i)  to  have  two  motions 
imparted  to  it  simultaneously,  one  of  which  would  carry  it  to  B 


POLYGON  OF  MOTIONS  15 

in  one  second,  and  the  other  to  C  in  one  second ;  and  that  it  is 
required  to  find  where  it  will  be  at  the  end  of  one  second,  and 
what  path  it  will  have  pursued.  c 

Imagine  the  body  to  move  in  obedience 
to  the  first  alone,  during  one  second  :  it 
would  thus  arrive  at  B ;  then  suppose  the 
second  motion  to  be  imparted  to  the  body, 
instead  of  the  first,  it  will  arrive  at  the  end  of  the  next  sec- 
ond at  D,  where  BD  is  equal  and  parallel  to  AC.  When 
the  two  motions  are  imparted  simultaneously,  instead  of  suc- 
cessively, the  same  point  D  will  be  reached  in  one  second, 
instead  of  two;  and  by  dividing  AB  and  AC  into  the  same 
(any)  number  of  equal  parts,  we  can  prove  that  the  body  will 
always  be  situated  at  some  point  of  the  diagonal  AD  of  the 
parallelogram,  hence  that  it  moves  along  AD.  Hence  follows 
the  proposition  known  as  the  parallelogram  of  motions. 

PARALLELOGRAM    OF    MOTIONS. 

If  there  be  simultaneously  impressed  on  a  body  two  velocities, 
which  would  separately  be  represented  by  the  lines  AB  and  AC, 
the  actual  velocity  will  be  represented  by  the  line  AD.  which  is 
the  diagonal  of  the  parallelogram  of  which  AB  and  AC  are  the 
adjacent  sides. 

§17.  Polygon  of  Motions.  —  In  all  the  above  cases,  the 
point  reached  by  the  body  at  the  end  of  a  second  when  the 
two  motions  take  place  simultaneously  is  the  same  as  that  which 
would  be  reached  at  the  end  of  two  seconds  if  the  motions  took 
place  successively ;  and  the  path  described  is  the  straight  line 
joining  the  initial  position  of  the  body,  with  its  position  at  the 
end  of  one  second  when  the  motions  are  simultaneous. 

The  same  principle  applies  whatever  be  the  number  of 
velocities  that  may  be  imparted  to  a  body  simultaneously. 
Thus,  if  we  suppose  the  several  velocities  imparted  to  be 
(Fig.  2)  AB,  AC,  AD,  AE,  and  AF,  and  it  be  required  to 


1 6  APPLIED   MECHANICS. 

determine  the  resultant  velocity,  we  first  let  the  body  move 
with  the  velocity  AB  for  one  second ;  at  the 
end  of  that  second  it  is  found  at  B ;  then  let 
it  move  with  the  velocity  AC  only,  and  "at 
the  end  of  another  second  it  will  be  found 
at  c ;  then  with  AD  only,  and  at  the  end  of 
the  third  second  it  will  be  found  at  d;  at  the 
end  of  the  fourth  at  e;  at  the  end  of  the  fifth 
at  /.  Hence  the  resultant  velocity,  when  all 
are  imparted  simultaneously,  is  Af,  or  "the 

closing  side  of  the  polygon. 

This  proposition  is  known  as  the  polygon  of  motions. 

POLYGON   OF    MOTIONS. 

If  there  be  simultaneously  impressed  on  a  body  any  number 
of  velocities,  the  resulting  velocity  will  be  represented  by  the 
closing  side  of  a  polygon  of  which  the  lines  representing  tJie 
separate  velocities  form  the  other  sides. 

§  1 8.  Characteristics  of  a  Force A  force  has  three 

characteristics,  which,  when  known,  determine  it ;  viz.,  Point 
of  Application,  Direction,  and  Magnitude.  These  can  be  repre- 
sented by  a  straight  line,  whose  length  is  made  proportional  to 
the  magnitude  of  the  force,  whose  direction  is  that  of  the 
motion  which  the  force  imparts,  or  tends  to  impart,  and  one  end 
of  which  is  the  point  of  application  of  the  force ;  an  arrow-head 
being  usually  employed  to  indicate  the  direction  in  which  the 
force  acts. 

§  19.   Parallelogram  of   Forces. 

PROPOSITION.  —  If  two  forces  acting  simultaneously  at  the 
same  point  be  represented,  in  point  of  application,  direction, 
and  magnitude,  by  two  adjacent  sides  of  a  parallelogram,  their 
resultant  will  be  represented  by  the  diagonal  of  the  parallelo- 
gram, drawn  from  the  point  of  application  of  the  two  forces. 

PROOF. — In  the  last  part  of  §   16  was  proved  the  propo* 


PARALLELOGRAM  OF  FORCES.  I/ 

sition  known  as  the  Parallelogram  of  Motions,  for  the  state- 
ment of  which  the  reader  is  referred  to  the  close  of  that 
section. 

We  have  also  seen  that  forces  are  proportional  to  the  velo- 
cities which  they  impart,  or  tend  to  impart,  in  a  unit  of  time, 
to  the  same  body. 

Hence  the  lines  representing  the  two  impressed  forces  are 
coincident  in  direction  with,  and  proportional  to,  the  lines  repre- 
senting the  velocities  they  would  impart  in  a  unit  of  time  to 
the  same  body ;  and  moreover,  since  the  resultant  velocity  is 
represented  by  the  diagonal  of  the  parallelogram  drawn  with 
the  component  velocities  as  sides,  the  resultant  force  must  coin- 
cide in  direction  with  the  resultant  velocity,  and  the  length  of 
the  line  representing  the  resultant  force  will  bear  to  the  result- 
ant velocity  the  same  ratio  that  one  of  the  component  forces 
bears  to  the  corresponding  velocity.  Hence  it  follows,  that  the 
resultant  force  will  be  represented  by  the  diagonal  of  the  paral 
lelogram  having  for  sides  the  two  component  forces. 

§  20.   Parallelogram  of   Forces :  Algebraic  Solution. 

PROBLEM.  —  Given  two  forces  F  and  F,  acting  at  the  same 
point  A  (Fig.  3),  and  inclined  to  each  other  at  an  angle  0  ;  required 
the  magnitude  and  direction  of  the  resultant 
force. 

Let  AC  represent  F,  AB  represent  Ft, 
and  let  angle  BAG  —  0  ;  then  will  R  =  AD    A 
represent  in  magnitude  and  direction   the 
resultant  force.     Also  let  angle  DAC  —  a;  then  from  the  tri~ 
angle  DAC  we  have 

AD2  =  AC2  +  CD2  -  2AC.  CDcosACD. 
But 

ACD  =  180°  -  0        .'.     cosACD  =  -cos* 

.'.     R2       =  F2  +  F2  +  2FF,  cos  (9 


+  F2  -f  2FF,  cos  (9. 


i8 


APPLIED  MECHANICS. 


This  determines  the  magnitude  of  R.  To  determine  its  direc- 
tion, let  angle  CAD  —  a.  .'.  angle  BAD  =  0  —  a,  and  we 
shall  have  from  the  triangle  DAC 


or 


and  similarly 


CD  :  AD  =  sin  CAD  :  smACD, 
Ft :  R  =  sin  a  :  sin  0 

T? 

.*.      sin  a     =  — -sin0, 
R 


sin(0-a)  =      sin0. 
R 


EXAMPLES. 


.  Given  F  =  47-34, 


75.46,  0  =     73°  14'  21";  find  R  and  a. 


2°.  Given  ^  =     5.36,  Fl  =    4.27,  0  =     32°  10'         ;  find  R  and  a. 
3°.  Given  F  =  42.00,  Ft  =  31.00,  0  =  150°  ;  find  R  and  a. 

4°.  Given  F  =  47.00,  Ft  —  75.00,  6  =  253°  ;  find  R  and  a. 


§21.  Parallelogram  of  Forces  when  6  =  90°.  —  When 
the  two  given  forces  are  at  right  angles  to  each  other,  the  for- 
mulae become  very  much  simplified,  since  the  parallelogram 
becomes  a  rectangle. 

From  Fig.  4  we  at  once  deduce 


R     =  V^F*  +  ^;», 

sin  a  =  — ^, 
R 

COS  a  =  —. 


1°.  Given  ^  = 

2°.  Given  ^  = 

3°.  Given  ^  = 

4°.  Given  /?  = 


3.0,  ^  = 

3.0,  Ft  = 

5.0,  Fl  = 

23.2,  Ft  = 


5.0  ;  find  ^  and  a. 

—  5.0  ;  find  i?  and  a. 

12.0  ;  find  ^  and  a. 

21.3  ;  find  R  and  a. 


DECOMPOSITION  OF  FORCES  IN  ONE    PLANE.  19 

§  22.  Triangle  of  Forces.  —  If  three  forces  be  represented* 
in  magnitude  and  direction,  by  the  three  sides  of  a  triangle  taken 
in  order,  then,  if  these  forces  be  simultaneously  applied  at  one 
point,  they  will  balance  each  other. 

Conversely,  three  forces  which,  when  simultaneously  applied 
at  one  point,  balance  each  other,  can  be  correctly  represented  in 
magnitude  and  direction  by  the  three  sides  of  a  triangle  taken  in 
order. 

These  propositions,  which  find  a  very  extensive  application, 
especially  in  the  determination  of  the  stresses  in  roof  and 
bridge  trusses,  are  proved  as  follows  :  — 

If  we  have  two  forces,  AC  and  AB  (see  Fig.  3),  acting  at  the 
point  A,  their  resultant  is,  as  we  have  already  seen,  AD ;  and 
hence  a  force  equal  in  magnitude  and  opposite  in  direction  to 
AD  will  balance  the  two  forces  AC  and  AB.  Now,  the  sides  of 
the  triangle  AC  DA,  if  taken  in  order,  represent  in  magnitude 
and  direction  the  force  AC,  the  force  CD  or  AB,  and  a  force 
equal  and  opposite  to  AD ;  and  these  three  forces,  if  applied  at 
the  same  point,  would  balance  each  other.  Hence  follows  the 
proposition. 

Moreover,  we  have 

AC  :  CD  \  DA  =  sinAUC     :  sin  CAD  :  smACZ>, 
or 

F-.F,     \R      =  sin(0  -a)  :  sin  a          :  sintf; 

or  each  force  is,  in  this  case,  proportional  to  the  sine  of  the 
angle  between  the  other  two. 

§  23.  Decomposition  of  Forces  in  one  Plane.  —  It  is 
often  convenient  to  resolve  a  force  into  two  components,  in  two 
given  directions  in  a  plane  containing  the  force.  Thus,  suppose 
we  have  the  force  R  =  AD  (Fig.  3),  and  we  wish  to  resolve  it 
into  two  components  acting  respectively  in  the  directions  AC 
and  AB ;  i.e.,  we  wish  to  find  two  forces  acting  respectively  in 
these  directions,  of  which  AD  shall  be  the  resultant :  we 


20 


APPLIED   MECHANICS. 


determine  these  components  graphically  by  drawing  a  parallelo- 
gram, of  which  AD  shall  be  the  diagonal,  and  whose  sides  shall 
have  the  directions  AC  and  AB  respectively.  The  algebraic 
values  of  the  magnitudes  of  the  compo- 
nents can  be  determined  by  solving  the 
triangle  ADC.  In  the  case  when  the 
directions  of  the  components  are  at  right 
angles  to  each  other,  let  the  force  R 
(Fig.  5),  applied  at  O,  make  an  angle  a 
with  OX.  We  may,  by  drawing  the  rect- 
angle shown  in  the  figure,  decompose  R 

into  two  components,  F  and  Fu  along  OX  and  O  Y  respectively ; 
and  we  shall  readily  obtain  from  the  figure, 


F  =  R  cos  a,     Fi  =  R  sin  a. 


FIG.  6. 


EXAMPLES. 

i°.  The  force  exerted  by  the  steam  upon  the  piston  of  a  steam-engine 

at  the  moment  when  it  is  in  the  position  shown  in  the  figure  is  AB  = 

1000   Ibs.      The   resistance   of  the 

guides  upon  the  cross-head  DE  is 

vertical.    Determine  the  force  acting 

along  the   connecting-rod  AC  and 

the   pressure   on   the   guides ;    also 

resolve  the  force  acting  along  the  connecting-rod  into 
two  components,  one   along,  and  the 
other  at  right  angles  to,  the  crank  OC. 
2°.  A  load  of  500  Ibs.  is  placed  at 
the  apex  C  of  the  frame  ACB ':   find 

the  stresses  in  AC  and  CB  respectively. 

3°.  A  load  of  4000  Ibs.  is  hung  at  C,  on  the  crane 

ABC:  find  the  pressure  in  the  boom  BC,  and  the  pull 

on  the  tie  AC,  where  BC  makes  an  angle  of  60°  with  the  horizontal, 

and  AC  an  angle  of  15°. 


COMPOSITION  OF  FORCES  IN  ONE   PLANE.  21 

4°.  A  force  whose  magnitude  is  7  is  resolved  into  two  forces  whose 
magnitudes  are  5  and  3  :  find  the  angles  they  make  with  the  given; 
force. 

§  24.  Composition  of  any  Number  of  Forces  in  One 
Plane,  all  applied  at  the  Same  Point. 

(a)  GRAPHICAL  SOLUTION.  —  Let  the  forces  be  represented 
(Fig.  2)  by  AB,  AC,  AD,  AE,  and  AF  respectively.     Draw  Be 
||  and  =  AC,  cd  ||  and  =  AD,  de  ||  and  =  AE,  and  ef  j|  and  = 
AF;    then  will  Af  represent  the  resultant  of  the  five  forces. 
This   solution   is   to   be   deduced    from 

§  17  in  the  same  way  as  §  19  is  deduced 
from  §  1  6.  c, 

(b)  ALGEBRAIC     SOLUTION.  —  Let 
the    given    forces    (Fig.    9),    of    which    B, 
three  are  represented  in  the  figure,  be 

F,  Ft)  F2,  Fy  F4,  etc.  ;  and  let  the  angles      l 

made  by  these  forces  with  the  axis  OX    o1^  <•»      £  —  jj 

be    a,   a,,    02,   a3,  a4,    etc.,    respectively.  FlG-9- 

Resolve   each    of   these   forces    into    two    components,  in   the 

directions  OX  and  OY  respectively.     We  shall  obtain  for  the 

components  along  OX 

OA  =  Fcosa,     OB   =  ^cosa,,     OC  —  F2cosa2,     etc.; 
and  for  those  along  OY 

OA,  =  Fs'ma,     OB,  ==  ^sino,,     OC,  =  J?2sma2,     etc. 


These  forces  are  equivalent  to  the  following  two  ;  viz.,  a 
force  Fcos  a  -f  F,  cos  a,  +  F2  cos  a2  -f-  F3  cos  a3  +  etc.  along  OX, 
and  a  force  .Fsin  a  +  F\  $m  ai  +  F*  siri  «2  +  Fz  sin  a3  -f-  etc.  along 
OY.  The  first  may  be  represented  by  ^Fcosa,  and  the  second 
by  ^Fsina,  where  2,  stands  for  algebraic  sum.  There  remains 
only  to  find  the  resultant  of  these  two,  the  magnitude  of  which 
is  given  by  the  equation 

R  =  V(2^cosa)2  -j- 


22 


APPLIED  MECHANICS. 


and,  if  we  denote  by  a^  the  angle  made  by  the  resultant  with 
OX,  we  shall  have 


COS  ar  = 


sin  OT 


R 


EXAMPLES. 


a3  =   112 


Find  the  result- 
ant force  and 
its  direction. 


Solution. 


F. 

a. 

COS  a. 

sin  a. 

F  COS  a. 

F  sin  a. 

47 

21° 

0-93358 

o.35837 

43.87826 

16.84339 

73 

48° 

0.66913 

o-743i5 

48.84649 

54-24995 

43 

82° 

O.I39I7 

0.99027 

5-98431 

42.58161 

23 

112° 

-0.37461 

0.92718 

-8.6l603 

21.32414 

90.09303 

134.99909 

*.  2^  cos  a  =  90.09303, 
-.  R 


a  =  134.99909, 


log  ^F  COS  a    =    1.954691 

=  2.210331 


+  (2F  sin  a)2  =  162.2976. 


log  COS  Or  =   9.744360 

Or  =   56°  I/- 

OBSERVATION.  —  It  would  be  perfectly  correct  to  use  the  minus  sign 
in  extracting  the  square  root,  or  to  call  R  =  —162.2976  ;  but  then  we 
should  have 


€050,=   90.09303 


or 


—  162.2976 
i8o°  -f  56° 


-  134.99909  ? 
—  162.2976 

=  236°  -  7'; 


COMPOSITION  OF  FORCES  APPLIED   AT  SAME   POINT.      2$ 

a  result  which,  if  plotted,  would  give  the  same  force  as  when  we  call 
R  =  162.2976     and     a*  ==  56°  if. 

Hence,  since  it  is  immaterial  whether  we  use  the  plus  or  the  minus  sign 
in  extracting  the  square  root  provided  the  rest  of  the  computation  be 
consistent  with  it,  we  shall,  for  convenience,  use  always  plus. 

*  =     77°> 


3°. 


a,=     82°, 
a2  =   163°, 

«S=    275°- 


a,  =  o, 

«2   =     90°. 


§25.  Polygon  of  Forces.  —  If  any  number  of  forces  be 
represented  in  magnitude  and  direction  by  the  sides  of  a  polygon 
taken  in  order,  then,  if  these  forces  be  simultaneously  applied  at 
one  point,  they  will  balance  each  other. 

Conversely,  any  number  of  forces  which,  when  simultaneously 
applied  at  one  point,  balance  each  other,  can  be  correctly  repre- 
sented in  magnitude  and  direction  by  the  sides  of  a  polygon  taken 
in  order. 

These  propositions  are  to  be  deduced  from  §  24  (a)  in  the 
same  way  as  the  triangle  of  forces  is  deduced  from  the  parallelo- 
gram of  forces. 

§  26.  Composition  of  Forces  all  applied  at  the  Same 
Point,  and  not  confined  to  One  Plane.  —  This  problem  can 
be  solved  by  the  polygon  of  forces,  since  there  is  nothing  in 
the  demonstration  of  that  proposition  that  limits  us  to  a  plane 
rather  than  to  a  gauche  polygon. 

The  following  method,  however,  enables  us  to  determine 
algebraic  values  for  the  magnitude  of  the  resultant  and  for  its 
direction. 


24 


APPLIED   MECHANICS. 


FIG.  10. 


We  first  assume  a  system  of  three  rectangular  axes,  OX, 

OY,  and  OZ  (Fig.  10),  whose  origin 
is  at  the  common  point  of  the  given 
forces.  Now,  let  OE  =  F  be  one 
of  the  given  forces.  First  resolve 
it  into  two  forces,  OC  and  OD,  the 
first  of  which  lies  in  the  z  axis,  and 
the  second  perpendicular  to  OZ, 
x  or,  as  it  is  usually  called,  in  the  z 
plane ;  the  plane  perpendicular  to 
OX  being  the  x  plane,  and  that 
perpendicular  to  OY  the  y  plane. 
Then  resolve  OD  into  two  com- 
ponents, OA  along  OX,  and  OB  along  OY.  We  thus  obtain 
three  forces,  OA,  OB,  and  OC  respectively,  which  are  equivalent 
to  the  single  force  OE.  These  three  components  are  the  edges 
of  a  rectangular  parallelepiped,  of  which  OE  =  Fis  the  diagonal. 
Let,  now, 

angle  EOX  =  a,     EOY  =  (3,     and     EOZ  =  y ; 

and  we  have,  from  the  right-angled  triangles  EOA,  EOB,  and 
EOC  respectively, 

OA  =  Fcosa,     OB  =  Fcosp,     OC  =  Fcosy. 
Moreover, 

OA2  +  OB2  =  OD2  and  OD2     +  OC2    =  OE2 
.'.     OA2     +  OB2     +  OC2    =  OE2, 

and  by  substituting  the  values  of  OA,  OB,  and  OC,  given  above, 
we  obtain 

COS2  a  -j-  COS2  (3  +  COS2  y  =   I  ; 

a  purely  geometrical  relation  existing  between  the  three  angles 
that  any  line  makes  with  three  rectangular  co-ordinate  axes. 

When  two  of  the  angles  a,  /3,  and  y  are  given,  the  third  can 
be  determined  from  the  above  equation. 


COMPOSITIOAr  OF- FORCES  APPLIED   AT  SAME   POINT.      2$ 


Resolve,  in  the  same  way,  each  of  the    given   forces   into 
three  components,  along  OX,  OY,  and  OZ  respectively,  and  we 
shall  thus  reduce  our  entire  system 
of   forces    to    the   following  three 
forces  :  — 


i°.  A  single  force  2/7  cos  a  along  OX. 
2°.  A  single  force  2/7 cos  ft  along  OY. 
3°.  A  single  force  2/7 cosy  along  OZ. 

We  next  proceed  to  find  a  sin- 
gle resultant  for  these  three  forces. 
Let  (Fig.  ii) 

OA  =  2/7  cos  a 
OB  =  2/7  cos  ft 
OC  = 


FIG.  xx. 


Compounding  OA  and  OB,  we  find  OD  to  be  their  resultant  ; 
and  this,  compounded  with  OC,  gives  OE  as  the  resultant  of 
the  entire  system.  Moreover, 

OE2  =  OD2  -4-  OC2  =  OA2  +  OB2  +  OC2, 
or 

fc      =  (2/7  cos  a)*  4-  (S^cos^)2  -h  (2/?  cosy)* 


(  2.F  cos  0) 


and  if  we  let  BOX  —  ar,  EOY  =  ft,  and 
have 


(2/7  cosy)*; 

=  yr,  we  shall 


cos  a"  = 


OA 
OE 


R 


2/7 cos  8  2/7  co 

r  = Y~^,   and  cosyr  =  ^— - 


This  gives  us  the  magnitude  and  direction  of  the  resultant. 

The  same  observation  applies  to  the  sign  of  the  radical  for 
R  as  in  the  case  of  forces  confined  to  one  plane. 


26 


APPLIED   MECHANICS. 


DETERMINATION   OF   THE   THIRD   ANGLE   FOR  ANY   ONE   FORCE. 

When  two  of  the  angles  a,  /3,  and  y  are  given,  the  cosine  of 
the  third  may  be  determined  from  the  equation,  — 

cos2  a  +  cos2/?  +  cos2  y  =  i  ; 

but,  as  we  may  use  either  the  plus  or  the  minus  sign  in  extract* 
ing  the  square  root,  we  have  no  means  of  knowing  which  of 
the  two  supplementary  angles  whose  cosine  has  been  deduced 
is  to  be  used. 

Thus,  suppose  a  =  45°,  (3  =  60°,  then 


cosy  =  ±i  --  i  -  J  =  ±f 
/.    y  =  60°,  or  1 20°  ; 

but  which  of  the  two  to  use  we  have  no  means  of  deciding. 

This  indetermination  will  be  more  clearly  seen  from  the  fol- 
lowing geometrical  considerations  :  — 

The  angle  a  (Fig.  12),  being  given   as  45°,  locates  the  line 

representing  the  force  on  a  right 
circular  cone,  whose  axis  is  OX, 
and  whose  semi-vertical  angle  is 
AOX-BOX  =  4$°.  On  the  other 
hand,  the  statement  that  (3  =  60° 
locates  the  force  on  another  right 
circular  cone,  having  O  Y  for  axis, 
and  a  semi-vertical  angle  of  60°; 
both  cones,  of  course,  having  their 
vertices  at  O.  Hence,  when  a  and 
(3  are  given,  we  know  that  the  line 

representing  the  force  is  an  element  of  both  cones  ;  and  this  is 
all  that  is  given. 

(a)  Now,  if  the  sum  of  the  two  given  angles  is  less  than 
90°,  the  cones  will  not  intersect,  and  the  data  are  consequently 
inconsistent. 


DETERMINATION  OF   THE    THIRD   ANGLE.  2/ 

(b)  If,  on  the  other  hand,  one  of  the  given  angles  being 
greater  than  90°,  their  difference  is  greater  than  90°,  the  cones 
will  not  intersect,  and  the  data  are  again  inconsistent. 

(c)  If  a  +  /?  =  90°,  the  cones  are  tangent  to  each  other, 
and  7  =  90°. 

(d)  If  a  -f-  J3  >  90°,  and  a  —  /?  or  /3  —  «<  90°,  the  cones 
intersect,  and  have  two  elements  in  common ;  and  we  have  no 
means  of  determining,  without  more  data,  which  intersection 
is  intended,  this  being  the  indetermination  that  arises  in  the 
algebraic  solution. 


I.  Given 


F  = 


EXAMPLES. 


63     a  =  53° 

49    a  =  87° 
2     £  =  70° 


ft  =  42' 
7  =  72' 
7  =  45' 


Find  the  magnitude 
and  direction  of 
the  resultant. 


Solution. 


p 

a. 

p. 

Y- 

COS  a. 

cosp. 

COSY. 

F  COS  a. 

/^COS/3. 

F  COS  y. 

63 

49 

2 

53° 
87° 

42° 
700 

72° 

45° 

0.60182 
0.05234 
0.6l888 

0.74314 
0.94961 
0.34202 

0.29250 
0.30902 
0.70711 

37.91466 
2.56466 
1.23776 

46.81782 
46.53089 
0.68404 

18.42750 
15.14198 
I.4I422 

41.71708 

2/^cos  a 

94-03275 
2/^cos  3 

34-98370 

2.F  cos  y 

R  =  V(S^cosa)2  -j-  (XF cos/3)2  +  (S/? cosy)2  =  108.6569. 

log  2/^cosa  =  1.620314       log  S^cos/^  =  1.973279       log  2/^cosy  =  1.543866 
log  j?  =  2.036057       log  R  =  2.036057       log  R  =  2.036057 


log  cos  ar      =9-584257       Iogcosj8r      =9.937222      log  cos  yr      =9.507809 
ar  =  67°  25'  20X/  (3r  =  30°  4'  i4/x    »  =71°  13'  5" 


28 


APPLIED   MECHANICS. 


F. 

a. 

0- 

F. 

a. 

£• 

V- 

2. 

4-3 

47°   2' 

65°  7' 

3- 

5 

9°° 

90° 

37.5 

88°  3' 

10°  5' 

7 

0° 

6.4 

68°  4' 

83°     2' 

4 

0° 

75 

73° 

45° 

§  27.  Conditions  of  Equilibrium  for  Forces  applied  at  a 
Single  Point. 

i°.  When  the  forces  are  not  confined  to  one  plane,  we  have 
already  found,  for  the  square  of  the  resultant, 


But  this  expression  can  reduce  to  zero  only  when  we  have 
a  =  o,     S/^cos  (3  =  o,     and     2/^cos  y  =  o  ; 


for  the  three  terms,  being  squares,  are  all  positive  quantities, 
and  hence  their  sum  can  reduce  to  zero  only  when  they  are 
separately  equal  to  zero. 

Hence  :  If  a  set  of  balanced  forces  applied  at  a  single  point 
be  resolved  into  components  along  three  directions  at  right  angles 
to  each  other,  the  algebraic  sum  of  the  components  of  the  forces 
along  each  of  the  three  directions  must  be  equal  to  zero,  and  con- 
versely. 

2°.  When  the  forces  are  all  confined  to  one  plane,  let  that 
plane  be  the  z  plane  ;  then  y  =  90°  in  each  case,  and 

/.      (3          =  90°  -  a 

/.     cos  (3  =  sin  a 

/.     fc       =  (^F  cos  a)2  4- 


Hence,  for  equilibrium  we  must  have 

cos  a)2  4-  CSJ?  sin  a)2  =  o; 


STATICS  OF  RIGID   BODIES.  29 

and,  since  this  is  the  sum  of  two  squares, 

—  o,  and  S/^sina  =  o. 


Hence  :  If  a  set  of  balanced  forces,  all  situated  in  one  plane  \ 
and  acting  at  one  point,  be  resolved  into  components  along  two 
directions  at  right  angles  to  each  other,  and  in  their  own  plane, 
the  algebraic  sum  of  the  components  along  each  of  the  tzvo  given 
directions  must  be  equal  to  zero  respectively;  and  conversely. 

§  28.  Statics  of  Rigid  Bodies.  —  A  rigid  body  is  one  that 
does  not  undergo  any  alteration  of  shape  when  subjected  to 
the  action  of  external  forces.  Strictly  speaking,  no  body  is 
absolutely  rigid  ;  but  different  bodies  possess  a  greater  or  less 
degree  of  rigidity  according  to  the  material  of  which  they  are 
composed,  and  to  other  circumstances.  When  a  force  is  ap- 
plied to  a  rigid  body,  we  may  have  as  the  result,  not  merely  a 
rectilinear  motion  in  the  direction'  of  the  force,  but,  as  will  be 
shown  later,  this  may  be  combined  with  a  rotary  motion  ;  in 
short,  the  criterion  by  which  we  determine  the  ensuing  motion 
is,  that  the  effect  of  the  force  will  distribute  itself  through  the 
body  in  such  a  way  as  not  to  interfere  with  its  rigidity. 

What  this  mode  of  distribution  is,  we  shall  discuss  here- 
after ;  but  we  shall  first  proceed  to  some  propositions  which  can 
be  proved  independently  of  this  consideration. 

§  29.  Principle  of  Rectilinear  Transferrence  of  Force  in 
Rigid  Bodies.  —  If  a  force  be  applied  to  a  rigid  body  at  the 
point  A  (Fig.  13)  in  the  direction  AB, 
whatever  be  the  motion  that  this  force 
would  produce,  it  will  be  prevented  from 
taking  place  if  an  equal  and  opposite 
force  be  applied  at  A,  B,  C,  or  D,  or  at  FlG-  I3- 

any  point  along  the  line  of  action  of  the  force  :  hence  we  have 
the  principle  that  — 

The  point  of  application  of  a  force  acting  on  a  rigid  body, 
may  be  transferred  to  any  other  point  which  lies  in  the  line  of 


APPLIED  MECHANICS. 


action  of  the  force,  and  also  in  the  body,  without  altering  the 
resulting  motion  of  the  body,  although  it  does  alter  its  state  of 
stress. 

§  30.  Composition  of  two  Forces  in  a  Plane  acting  at 
Different  Points  of  a  Rigid  Body,  and  not  Parallel  to  Each 
Other.  —  Suppose  the  force  F  (Fig.  14)  to  be  applied  at  A,  and 
F,  at  Bt  both  in  the  plane  of  the  paper,  and  acting  on  the  rigid 
body  abcdef.  Produce  the  lines  of  direction  of  the  forces  till 
they  meet  at  <9,  and  suppose  both  F  and  F,  to  act  at  O.  Con- 
struct the  parallelogram  ODHE,  where  OD  =  F  and  OE  =  Ft ; 

then  will  OH  —  R  rep^ 
resent  the  resultant 
force  in  magnitude  and 
in  direction.  Its  point 
of  application  may  be 
conceived  at  any  point 
along  the  line  OH,  as 


at  C,  or  any  other 
point  ;  and  a  force 
equal  and  opposite  to 
OH,  applied  at  any  point  of  the  line  OH,  will  balance  F  at  A, 
and  F,  at  B. 

The  above  reasoning  has  assumed  the  points  A,  B,  C  and 
O,  all  within  the  body :  but,  since  we  have  shown,  that  when 
this  is  the  case,  a  force  equal  and  opposite  to  R  at  C  will  bal- 
ance Fat  A,  and  Ft  at  B,  it  follows,  that  were  these  three  forces 
applied,  equilibrium  would  still  subsist  if  we  were  to  remove 
the  part  bafeghc  of  the  rigid  body  ;  or,  in  other  words,  — 

The  same  construction  holds  even  when  the  point  O  falls  out- 
side the  rigid  body. 

§31.  Moment  of  a  Force  with  Respect  to  an  Axis  Per- 
pendicular to  the  Force. 

DEFINITION.  —  The  moment  of  a  force  with  respect  to  an 
axis  perpendicular  to  the  force,  and  not  intersecting  it,  is  the 


FIG.  14. 


EQUILIBRIUM  OF   THREE   PARALLEL   FORCES. 


FIG.  15. 


product  'of  the  force  by  the  common  perpendicular  to  (shortest 
distance  between)  the  force  and  the  axis. 

Thus,  in  Fig.  15  the  moment  of  F  about 
an  axis  through  O  and  perpendicular  to  the 
plane  of  the  paper  is  F(OA).  The  sign  of 
the  moment  will  depend  on  the  sign  attached 
to  the  force  and  that  attached  to  the  perpen- 
dicular. These  will  be  assumed  in  this  book 
in  such  a  manner  as  to  render  the  following  true  ;  viz.,  — 

The  moment  of  a  force  with  respect  to  an  axis  is  called  posi- 
tive when,  if  the  axis  were  supposed  fixed,  the  force  would  cause 
the  body  on  which  it  acts  to  rotate  around  the  axis  in  the  direc- 
tion of  the  hands  of  a  watch  as 
seen  by  the  observer  looking  at 
the  face.  It  will  be  called  nega- 
tive when  the  rotation  would  take 
place  in  the  opposite  direction. 

§  32.  Equilibrium  of  Three 
Parallel  Forces  applied  at 
Different  Points  of  a  Rigid 
Body.  —  Let  it  be  required  to 
find  a  force  (Fig.  16)  that  will 
balance  the  two  forces  F  at  A, 
and  -Ft  at  B.  Apply  at  A  and  B 
respectively,  and  in  the  line  AB, 
the  equal  and  opposite  forces  Aa 
and  Bb.  Their  introduction  will 
produce  no  alteration  in  the 
body's  motion. 

The  resultant  of  F  and  Aa 
is  Af,  that  of  F,  and  Bb  is  Bg. 
Compound  these  by  the  method 
of  §  30,  and  we  obtain  as  result- 
ant ce.     A  force  equal  in  magnitude  and  opposite  in  direction 


FIG.  16. 


32  APPLIED   MECHANICS. 

to  cej  applied  at  any  point  of  the  line  cC,  will  be  the  force 
required  to  balance  Fat  A  and  F,  at  B  ;  and,  as  is  evident  from 
the  construction,  this  line  is  in  the  plane  of  the  two  forces. 
Moreover,  by  drawing  triangle  fKl  equal  to  Bbg,  we  can  readily 
prove  that  triangles  oce  and  Afl  are  equal  :  hence  the  angle  oce 
equals  the  angle  fAl,  and  R  is  parallel  to  /^and  Ft.  Also 

R  =  ce  =  ch  +  he  =  ,4  AT  +  A7  =  F  +  ^ 


__          _ 

AC       fK       Ad 
and 

CL~.  M.  =-  -5.- 

BC~  Bb~  Bb'y 
.'.    since  ^4#  =  ^ 


BC       F   "  BC       AC          AB 

where  qr  is  any  line  passing  through  C. 

Hence  we  have  the  following  propositions  ;  viz.,  — 
If  three  parallel  forces  balance  each  other,  — 
1°.    They  must  lie  in  one  plane. 

2°.  The  middle  one  must  be  equal  in  magnitude  and  opposite 
in  direction  to  the  sum  of  the  other 
two. 

3°.  Each  force  is  proportional  to  the 

fcj  IB  o  distance  between  the  lines  of  direction 
of  the  other  two  as  measured  on  any 
line  intersecting  all  of  them. 

The  third  of  the  above-stated  con- 
ditions may  be  otherwise  expressed, 
thus  :  — 

FIG.  17.  The  algebraic  sum  of  the  moments 

of  the  three  forces  about  any  axis  perpendicular  to  the  forces 
must  be  zero. 


RESULTANT  OF  A    PAIR   OF  PARALLEL   FORCES.  33 

PROOF.  —  Let  F,  Fa  and  R  (Fig.  17)  be  the  forces  ;  and  let 
the  axis  referred  to  pass  through  O.  Draw  OA  perpendicular 
to  the  forces.  Then  we  have 

F(OA)  +  Ft  (OB)  =  F(OC  +  CA)  +  Ft(OC  -  BC) 
=  (F+Fl)OC     +  F(AC)  - 

But,  from  what  we  have  already  seen, 

F  +  F,  =  -R 

and 

JL^JH 

BC       AC 

.-.     F(AC)  =  ^(^C) 
.-.    F(OA)  +  Ft(OB)  =  -R(OC)  -f  o 

F,(OB)  +  tf(0C)  =  o, 


or  the  algebraic  sum  of  the  moments  of  t\\Q  forces  about  the 
axis  through  O  is  equal  to  zero. 

§  33.  Resultant  of  a  Pair  of  Parallel  Forces.  —  In  the 
preceding  case,  the  resultant  of  any  two  of  the  three  forces 
Fy  Fiy  and  R,  in  Fig.  16  or  Fig.  17,  is  equal  and  opposite  to  the 
third  force.  Hence  follow  the  two  propositions  :  — 

I.  If  two  parallel  forces   act   in   the  same  direction,  their 
resultant  lies  in  the  plane  of  the  forces,  is  equal  to  their  sum, 
acts  in  the  same  direction,  and  cuts  the  line  joining  their  points 
of  application,  or  any  common  perpendicular  to  the  two  forces, 
at  a  point  which  divides  it  internally  into  two  segments  in- 
versely as  the  forces. 

II.  If  two  unequal  parallel  forces  act  in  opposite  directions, 
their  resultant  lies  in  the  plane  of  the  forces,  is  equal  to  their 
difference,  acts  in  the  direction  of  the  larger  force,  and  cuts  the 
line  joining  their  points  of  application,  or  any  common  perpen- 
dicular to  them,  at  a  point  which  (lying  nearer  the  larger  force) 


34  APPLIED   MECHANICS. 

divides  it  externally  into  two  segments  which  are  inversely  as 
the  forces. 

Another  mode  of  stating  the  above  is  as  follows  :  — 

i°.  The  resultant  of  a  pair  of  parallel  forces  lies  in  the  plane 
of  the  forces. 

2°.  It  is  equal  in  magnitude  to  their  algebraic  sum,  and  coin- 
cides in  direction  with  the  larger  force. 

3°.  The  moment  of  the  resultant  about  an  axis  perpendicu- 
lar to  the  plane  of  the  forces  is  equal  to  the  algebraic  sum  of 
the  moments  about  the  same  axis. 

EXAMPLES. 

1.  Find  the  length  of  each  arm  of  a  balance  such  that  i  ounce  at 
the  end  of  the  long  arm  shall  balance  i  pound  at  the  end  of  the  short 
arm,  the  length  of  beam  being  2  feet,  and  the  balance  being  so  propor- 
tioned as  to  hang  horizontally  when  unloaded. 

2.  Given  beam  =28  inches,  3  ounces  to  balance  15. 

3.  Given  beam  =  36  inches,  5  ounces  to  balance  25  ounces. 


MODE  OF  DETERMINING  THE  RESULTANT  OF  A  PAIR  OF  PARALLEL 
FORCES  REFERRED  TO  A  SYSTEM  OF  THREE  RECTANGULAR 
AXES. 

Let  both  forces  (Fig.  18)  be  parallel  to  OZ  '  ;  then  we  have, 
from  what  has  preceded, 

F  =  £  =  F±_F>  = 
be       ab  ac  a 

But  from  the  figure 


or 


.'.     Fx2  —  Fxt  =  FjX   —  F^2 


RESULTANT  OF  NUMBER   OF  PARALLEL   FORCES.          35 


and  similarly  we  may  prove  that 


or 


i°.  The  resultant  of  two  parallel  forces  is  parallel  to  the 
forces  and  equal  to  their  algebraic  sum. 


R=F+F, 


FIG.  18. 

2°.  The  moment  of  the  resultant  with  respect  to  OX  is 
equal  to  the  algebraic  sum  of  their  moments  with  respect  to 
OX ;  and  likewise  when  the  moments  are  taken  with  respect 
to  OY. 

§34.  Resultant  of  any  Number  of  Parallel  Forces. — 
Let  it  be  required  to  find  the  resultant  of  any  number  of  paral- 
lel forces. 

In  any  such  case,  we  might  begin  by  compounding  two  of 
them,  and  then  compounding  the  resultant  of  these  two  with  a 
third,  this  new  resultant  with  a  fourth,  and  so  on.  Hence,  for 
the  magnitude  of  any  one  of  these  resultants,  we  simply  add 
to  the  preceding  resultant  another  one  of  the  forces  ;  and  for 
the  moment  about  any  axis  perpendicular  to  the  forces,  we  add 


APPLIED   MECHANICS. 


to  the  moment  of  the  preceding  resultant  the  moment  of  the 
new  force. 

Hence  we  have  the  following  facts  in  regard  to  the  resultant 
of  the  entire  system  :  — 

I  °.  The  resultant  will  be  parallel  to  the  forces  and  equal  to 
their  algebraic  sum. 

2°.  The  moment  of  the  resultant  about  any  axis  perpendicular 
to  the  forces  will  be  equal  to  the  algebraic  sum  of  the  moments 
of  the  forces  about  the  same  axis. 

The  above  principles  enable  us  to  determine  the  resultant 
in  all  cases,  except  when  the  algebraic  sum  of  the  forces  is 
equal  to  zero.  This  case  will  be  considered  later. 

§  35.  Composition  of  any  System  of  Parallel  Forces 
Y  when  all  are  in  One  Plane. — 
Refer  the  forces  to  a  pair  of  rect- 
angular axes,  OX,  OY  (Fig.  19), 
and  assume  OY  parallel  to  the 
forces. 

The  forces  and  the  co-ordinates 
of  their  lines  of  direction  being  as 
indicated  in  the  figure,  if  we  denote 
0  by  R  the  resultant,  and  by  XQ  the 
co-ordinate  of  its  line  of  direction, 
we  shall  have,  from  the  preceding, 

R  =  ^F;  ( i ) 

and  if  moments  be  taken  about  an 
axis  through  O,  and  perpendicular 


F,       F, 


FIG.  19. 


to  the  plane  of  the  forces,  we  shall  also  have 

Rx0  =  -S.FX. 
Hence 

R  =  ^F     and     x(t  = 


(2) 


determine  the  resultant  in   magnitude   and   in  line  of   action, 
.except  when  %F  =  o,  which  case  will  be  considered  later. 


EQUILIBRIUM  OF  ANY  SET  OF  PARALLEL   FORCES.       $? 

§  36,  Composition  of  any  System  of  Parallel  Forces  not 
confined  to  One  Plane.  —  Refer  the  forces  to  a  set  of  rect- 
angular axes  so  chosen  that  OZ  is  parallel  to  their  direction. 
If  we  denote  the  forces  by  Fiy  F2,  Fy  F4,  etc.,  and  the  co-ordinates 
of  their  lines  of  direction  by  (*•„  7,),  (x2J  jj>2),  etc.,  and  if  we 
denote  their  resultant  by  R,  and  the  co-ordinates  of  its  line  of 
direction  by  (xm  j^0),  we  shall  have,  in  accordance  with  what  has 
been  proved  in  §  34,  — 

1°.  The  magnitude  of  the  resultant  is  equal  to  the  algebraic 
mm  of  the  forces  •,  or 

R  =  2F. 

2°.  The  moment  of  the  resultant  about  OY  is  equal  to  the 
mm  of  the  moments  of  the  forces  about  OY,  or 


3°.    The  moment  of  the  resultant  about  OX  is  equal  to  the 
of  the  moments  about  OX,  or 


Hence 


determine  the  resultant  in  all  cases,  except  when  2<F  =  o. 

§  37.  Conditions  of  Equilibrium  of  any  Set  of  Parallel 
Ferces.  —  If  the  axes  be  assumed  as  before,  so  that  OZ  is 
parallel  to  the  forces,  we  must  have 

^F  =  o,     ^Fx  =  o,     and     ^Fy  —  o. 


To  prove  this,  compound  all  but  one  of  the  forces.  Then  equilib- 
rium will  subsist  only  when  the  resultant  thus  obtained  is  equal 
and  directly  opposed  to  the  remaining  force  ;  i.e,  it  must  be 
equal,  and  act  along  the  same  line  and  in  the  opposite  direction. 
Hence,  calling  Ra  the  resultant  above  referred  to,  and  (xa,  ya) 
the  co-ordinates  of  its  line  of  direction,  and  calling  FH  the 


38  APPLIED   MECHANICS. 

remaining  force,  and  (xw  y^  the  co-ordinates  of  its  line  of  direc- 
tion, we  must  have 

Ra  =    ~Fn,      *a  =   *n,       J^  =  JK«, 

.  '  .      Ra  +   Fn  =         O,          RaXa  +   FnXn  —   O,          ^JFa  +  Fnyn  =   O, 

.-.    2F  =      o,       *ZFx  =o,       ^Fy  =  o. 


When  the  forces  are  all  in  one  plane,  the  conditions  become 
2F  =  o,     ^Fx  =  o. 

§  38.  Centre    of    a    System    of   Parallel    Forces.  —  The 

resultant  of  the  two  parallel  forces  F  and  Ft  (Fig.  20),  ap- 
plied at  A  and  B  respectively,  is  a  force  R  •=  F  -\-  Flt  whose 
line  of  action  cuts  the  line  AB  at  a  point  C, 
which  divides  it  into  two  segments  inversely  as 
the  forces.  If  the  forces  F  and  F,  are  turned 
through  the  same  angle,  and  assume  the  posi- 
tions AO  and  BOl  respectively,  the  line  of 
action  of  the  resultant  will  still  pass  through 
C,  which  is  called  the  centre  of  the  two  parallel 
forces  F  and  /v  Inasmuch  as  a  similar  reasoning  will  apply 
in  the  case  of  any  number  of  parallel  forces,  we  may  give  the 
following  definition  :  — 

The  centre  of  a  system  of  parallel  forces  is  the  point  through 
which  the  line  of  action  of  the  resultant  always  passes,  no  matter 
how  the  forces  are  turned,  provided  only  — 

i°.    Their  points  of  application  remain  the  same. 
2°.    Their  relative  magnitudes  are  unchanged. 
3°.   They  remain  parallel  to  each  other. 

Hence,  in  finding  the  centre  of  a  set  of  parallel  forces,  we 
may  suppose  the  forces  turned  through  any  angle  whatever,  and 
the  centre  of  the  set  is  the  point  through  which  the  line  of 
action  of  the  resultant  always  passes. 


DISTRIBUTED  FORCES. 


39 


§  39-  Co-ordinates  of  the  Centre  of  a  Set  of  Parallel 
Forces.  —  Let  Fl  (Fig.  21)  be  one  of  the  forces,  and  (xltyu  zj 
the  co-ordinates  of  its  point 
of  application.  Let  F2  be 
another,  and  (x2t  y2t  z2)  co- 
ordinates of  its  point  of 
application.  Turn  all  the 
forces  around  till  they  are 
parallel  to  OZ,  and  find  the 
line  of  direction  of  the  re- 
sultant force  when  they  are 
in  this  position.  The  co- 
ordinates of  this  line  are 

FIG.  21. 


and,  since  the  centre  of  the  system  is  a  point  on  this  line,  the 
above  are  two  of  the  co-ordinates  of  the  centre.  Then  turn 
the  forces  parallel  to  OX,  and  determine  the  line  of  action  of 
the  resultant.  We  shall  have  for  its  co-ordinates 


y*  = 


Hence,  for  the  co-ordinates  of  the  centre  of  the  system,  we 
have 


y0  = 


When  2F  =  o  the  co-ordinates  would  be  oo,  therefore  such 
a  system  has  no  centre. 

§  40.  Distributed  Forces While  we  have  thus  far  as- 
sumed our  forces  as  acting  at  single  points,  no  force  really  acts 
at  a  single  point,  but  all  are  distributed  over  a  certain  surface 


40  APPLIED   MECHANICS. 

or  through  a  certain  volume ;  nevertheless,  the  propositions 
already  proved  are  all  applicable  to  the  resultants  of  these 
distributed  forces.  We  shall  proceed  to '  discuss  distributed 
forces  only  when  all  the  elements  of  the  distributed  force  are 
parallel  to  each  other.  As  a  very  important  example  of  such  a 
distributed  force,  we  may  mention  the  force  of  gravity  which 
is  distributed  through  the  mass  of  the  body  on  which  it  acts. 
Thus,  the  weight  of  a  body  is  the  resultant  of  the  weights  of 
the  separate  parts  or  particles  of  which  it  is  composed.  As 
another  example  we  have  the  following :  if  a  straight  rod  be 
subjected  to  a  direct  pull  in  the  direction  of  its  length,  and  if 
it  be  conceived  to  be  divided  into  two  parts  by  a  plane  cross- 
section,  the  stress  acting  at  this  section  is  distributed  over  the 
surface  of  the  section. 

§41.  Intensity  of  a  Distributed  Force.  —  Whenever  we 
have  a  force  uniformly  distributed  over  a  certain  area,  we  obtain 
its  intensity  by  dividing  its  total  amount  by  the  area  over  which 
it  acts,  thus  obtaining  the  amount  per  unit  of  area. 

If  the  force  be  not  uniformly  distributed,  or  if  the  intensity 
vary  at  different  points,  we  must  adopt  the  following  means 
for  rinding  its  intensity.  Assume  a  small  area  containing  the 
point  under  consideration,  and  divide  the  total  amount  of  force 
that  acts  on  this  small  area  by  the  area,  thus  obtaining  the 
mean  intensity  over  this  small  area :  this  will  be  an  approxima- 
tion to  the  intensity  at  the  given  point ;  and  the  intensity  is  the 
limit  of  the  ratio  obtained  by  making  the  division,  as  the  area 
used  becomes  smaller  and  smaller. 

Thus,  also,  the  intensity,  at  a  given  point,  of  a  force  which 
is  distributed  through  a  certain  volume,  is  the  limit  of  the 
ratio  of  the  force  acting  on  a  small  volume  containing  the 
given  point,  to  the  volume,  as  the  latter  becomes  smaller  and 
smaller. 

§42.  Resultant  of  a  Distributed  Force. —  i°.  Let  the 
force  be  distributed  over  the  straight  line  AB  (Fig.  22),  and 


RESULTANT  OF  A   DISTRIBUTED   FORCE. 


let  its  intensity  at  the  point  E  where  AE  =  x,  be  represented 
by  EF '  =  p  —  <£(*•),  a  function  of  x ; 
then  will  the  force  acting  on  the  por- 
tion Ee  =  A^r  of  the  line  be/A^r:  and 
if  we  denote  by  R  the  magnitude  of 
the  resultant  of  the  force  acting  on  the 
entire  line  AB,  and  by  x0  the  distance 
of  its  point  of  application  from  A,  we  shall  have 

R  =  3/A.x  approximately, 
or 

R  =  fpdx  exactly ; 

and,  by  taking  moments  about  an  axis  through  A  perpendicular 
to  the  plane  of  the  force,  we  shall  have 

XoR  =  ^x(pkx)  approximately, 
or 

x0R  =  fpxdx  exactly ; 

whence  we  have  the  equations 


R  =  fpdx, 


Spdx 


Let  the  force  be  distributed  over  a  plane  area  EFGff 

(Fig.  23),  let  this  area  be  re- 
ferred to  a  pair  of  rectangular 
axes  OX  and  OV,  in  its  own 
plane,  and  let  the  intensity 
of  the  force  per  unit  of  area 
at  the  point  P,  whose  co- 
B  ordinates  are  x  and  y,  be 
p  =  $(x,  y) ;  then  will  p&x&y 
be  approximately  the  force  act- 
ing on  the  small  rectangular 
area  A^rAj/.  Then,  if  we  rep- 
resent by  R  the  magnitude  of 
the  resultant  of  the  distributed  force,  and  by  xmym  the  co-ordi- 


DC 
FIG.  23. 


42  APPLIED   MECHANICS. 

nates  of  the  point  at  which  the  line  of  action  of  the  resultant 
cuts  the  plane  of  EFGH,  we  shall  have 

R  —  2/A^cAy  approximately, 
x0R  = 


or,  as  exact  equations,  we  shall  have 

R  =  fSpdxdy, 

ffpxdxdy  =  ffpydxdy 

~  '  ~ 


3°.  Let  the  force  be  distributed  through  a  volume,  let  this 
volume  be  referred  to  a  system  of  rectangular  axes,  OX,  O  Y, 
and  OZ,  let  A  V  represent  the  elementary  volume,  whose  co- 
ordinates are  x,  y,  zt  and  let  p  =  <j>(x,  yy  z]  be  the  intensity  of 
the  force  per  unit  of  volume  at  the  point  (x,  y,  z)  ;  then,  if  we 
represent  by  R  the  magnitude  of  the  resultant,  and  by  x0,  y0,  zm 
the  co-ordinates  of  the  centre  of  the  distributed  force,  we  shall 
have,  from  the  principles  explained  in  §  38  and  §  39,  the  approx- 
imate equations 

R  = 


and  these  give,  on  passing  to  the  limit,  the  exact  equations 


R  -  MV          -  -  SpydV          - 

~Jpd     x°~'  y°~'     °~ 


§  43.  Centre  of'  Gravity.  —  The  weight  of  a  body,  or  system 
of  bodies,  is  the  resultant  of  the  weight  of  the  separate  parts 
or  particles  into  which  it  may  be  conceived  to  be  divided  ;  and 
the  centre  of  gravity  of  the  body,  or  system  of  bodies,  is  the 
centre  of  the  above-stated  system  of  parallel  forces,  i.e.,  the 
point  through  which  the  resultant  always  passes,  no  matter  how 
the  forces  are  turned.  The  weight  of  any  one  particle  is  the 
force  which  gravity  exerts  on  that  particle  :  hence,  if  we  repre- 


FORCE   APPLIED    TO    CENTRE   OF  STRAIGHT  ROD.         43 

sent  the  weight  per  unit  of  volume  of  a  body,  whether  it  be 
the  same  for  all  parts  or  not,  by  w,  we  shall  have,  as  an 
approximation, 


and  as  exact  equations, 


fwxdV  fwydV  fwzdV 

>    (0 


where  W  denotes  the  entire  weight  of  the  body,  and  xot  ym  z0, 
the  co-ordinates  of  its  centre  of  gravity. 

If,  on  the  other  hand,  we  let  M  =  entire  mass  of  the  body, 
dM  —  mass  of  volume  dVt  and  m  =  mass  of  unit  of  volume, 
we  shall  have 

W  =  Mg,     w  =  mg,     wdV  —mgdV  =  gdM. 
Hence  the  above  equations  reduce  to 

fxdM  fydM  fzdM 


Equations  (i)  and  (2)  are  both  suitable  for  determining  the 
centre  of  gravity;  one  of  the  sets  being  sometimes  most  con- 
venient, and  sometimes  the  other. 

§  44.  Centre  of  Gravity  of  Homogeneous  Bodies  __  If 
the  body  whose  centre  of  gravity  we  are  seeking  is  homogeneous, 
or  of  the  same  weight  per  unit  of  volume  throughout,  we  shall 
have,  that  w  ==.  a  constant  in  equations  (i)  ;  and  hence  these 
reduce  to 


§45.  Effect  of  a  Single  Force  applied  at  the  Centre  of 
a  Straight  Rod  of  Uniform  Section  and  Material.  —  If  a 
straight  rod  of  uniform  section  and  material  have  imparted  to  it 


44  APPLIED   MECHANICS. 

a  motion,  such  that  the  velocity  imparted  ima  unit  of  time  to 
each  particle  of  the  rod  is  the  same,  and  if  we  represent  this 
velocity  by/,  then  if  at  each  point  of  the  rod,  we  lay  off  a  line 
xy  (Fig.  24)   in   the  direction  of   the  motion, 
and  representing  the  velocity  imparted  to  that 
point,   the   line   bounding   the    other   ends    of 
the  lines  xy  will  be  straight,  and  parallel  to  the 
rod.     If  we   conceive  the   rod   to    be   divided 
into   any   number  of   small    equal    parts,   and 


denote  the  mass  of  one  of  these  parts  by  <\M,  then  will 
contain  as  many  units  of  momentum  as  there  are  units  of  force 
in  the  force  required  to  impart  to  this  particle  the  velocity 
f  in  a  unit  of  time  ;  and  hence  f&M  is  the  measure  of  this 
force. 

Hence  the  resultant  of  the  forces  which  impart  the  velocity 
f  to  every  particle  of  the  rod  will  have  for  its  measure 

fM, 

where  M  is  the  entire  mass  of  the  rod  ;  and  its  point  of  applica- 
tion will  evidently  be  at  the  middle  of  the  rod. 

It  therefore  follows  that  — 

The  effect  of  a  single  force  applied  at  the  middle  of  a  straight 
rod  of  uniform  section  and  material  is  to  impart  to  the  rod  a 
motion  of  translation  in  the  direction  of  the  force,  all  points  of  , 
the  rod  acquiring  equal  velocities  in  equal  times. 

§46.  Translation  and  Rotation  combined.  —  Suppose  that 
we.have  a  straight  rod  AB  (Fig.  25),  and  suppose  that  such  a 
force  or  such  forces  are  applied  to  it  as  will  impart  to  the  point 
A  in  a  unit  of  time  the  velocity  Aa,  and  to  the  point  B  the 
(different)  velocity  Bb  in  a  unit  of  time,  both  being  perpendicu- 
lar to  the  length  of  the  rod.  It  is  required  to  determine  the 
motion  of  any  other  point  of  the  rod  and  that  of  the  entire 
rod. 


TRANSLATION  AND   ROTATION  COMBINED. 


45 


FIG.  25. 


Lay  off  Aa  and  Bb  (Fig.  25),  and  draw  the  line  abt  and  pro- 
duce it  till  it  meets  AB  produced 
in  O :  then,  when  these  velocities 
Aa  and  Bb  are  imparted  to  the 
points  A  and  B,  the  rod  is  in  the 
act  of  rotating  around  an  axis 
through  O  perpendicular  to  the  plane  of  the  paper ;  for  when  a 
body  is  rotating  around  an  axis,  the  linear  velocity  of  any  point 
of  the  body  is  perpendicular  to  the  line  joining  the  point  in 
question  with  the  axis  (i.e.,  the  perpendicular  dropped  from  the 
point  in  question  upon  the  axis),  and  proportional  to  the  dis- 
tance of  the  point  from  the  axis. 

Hence  :  If  the  velocities  of  two  of  the  points  in  the  rod  are 
given,  and  if  these  are  perpendicular  to  the  rod,  the  motion 
of  the  rod  is  fixed,  and  consists  of  a  rotation  about  some  axis 
at  right  angles  to  the  rod. 

Another  way  of  considering  this  motion  is  as  follows  :  Sup- 
pose, as  before,  the  velocities  of  the  points  A  and  B  to  be 

represented  by  Aa  and  Bb  respec- 
tively, and  hence  the  velocity  of 
any  other  point,  as  x  (Fig.  26),  to 
be  represented  by  xy,  or  the  length 
of  the  line  drawn  perpendicular  to 
FIG.  26.  AB,  and  limited  by  AB  and  ab. 

Then,  if  we  lay  off  Aa,  —  Bb,  =  \(Aa  +  Bb)  =  Cc,  and  draw 
a,b,,  and  if  we  also  lay  off  Aa2  —  a,a,  and  Bb2  =  bjb,  we  shall 
have  the  following  relations  ;  viz.,  — 

Aa  =  Aa,  —  Aa2, 
Bb  =  Bb,  +  Bb2, 
xy  =  Xy,  —  xy2,  etc., 

or  we  may  say  that  the  actual  motion  imparted  to  the  rod  in  a 
unit  of  time  may  be  considered  to  consist  of  the  following  two 
parts :  — 


46  APPLIED   MECHANICS. 

i°.  A  velocity  of  translation  represented  by  AaJ}  the  mean 
velocity  of  the  rod ;  all  points  moving  with  this  velocity. 

2°.  A  varying  velocity,  different  for  every  different  point, 
and  such  that  its  amount  is  proportional  to  its  distance  from 
Cy  the  centre  of  the  rod,  as  graphically  shown  in  the  triangles 
Aa2CBb2.  In  other  words,  the  rod  has  imparted  to  it  two 
motions  :  — 

i°.  A  translation  with  the  mean  velocity  of  the  rod. 

2°.  A  rotation  of  the  rod  about  its  centre. 

§47.  Effect  of  a  Force  applied  to  a  Straight  Rod  of 
Uniform  Section  and  Material,  not  at  its  Centre.  —  If  the 
force  be  not  at  right  angles  to  the  rod,  resolve  it  into  two  com- 
ponents, one  acting  along  the  rod,  and  the  other  at  right  angles 
to  it.  The  first  component  evidently  produces  merely  a  trans- 
lation of  the  rod  in  the  direction  of  its  length  :  hence  the  second 
component  is  the  only  one  whose  effect  we  need  to  study. 

To  do  this  we  shall  proceed  to  show,  that,  when  such  a  rod 
has  imparted  to  it  the  motion  described  in  §  46,  the  single  re- 
A  cd  B  sultant  force  which  is  required  to  impart 

this  motion  in  a  unit  of  time  is  a  force 
acting  at  right  angles  to  the  rod,  at  a  point 
different  from  its  centre ;  and  we  shall  de- 

(/ 

FIG.  27.  termine  the  relation  between  the  force  and 

the  motion  imparted,  so  that  one  may  be  deduced  from  the 
other. 

Let  A  be  the  origin  (Fig.  27),  and  let 

Ac    =  x,  cd  =  dx. 

AB  —  I  =±  length  of  the  rod. 

ce  =f=  velocity  imparted  per  unit  of  time  at  distance  x 
from  A. 

Aa  =  /„  Bb  =  f2. 

w     —  weight  per  unit  of  length. 

m     —  mass  per  unit  of   length  =  ^!. 

g 


EFFECT  OF  FORCE   APPLIED    TO   A   STRAIGHT  ROD.       47 

W    =  entire  weight  of  rod. 

M    =  entire  mass  of  rod  —  —  . 

g 
R     =  single  resultant  force  acting  for  a  unit  of  time  to 

produce  the  motion. 
x0     —  distance  from  A  to  point  of  application  of  R. 

Then  we  shall  have, 


Hence,  from  §  42, 

AabB)  =  ^(/  +/,)/  =  ^(/  +  /2).  (i) 

2  2 

(2) 


.        I  /.  +  '/.  ,  (-) 

"  3  /,  +/, 

We  thus  have  a  force  R,  perpendicular  to  AB,  whose  mag- 
nitude is  given  by  equation  (i),  and  whose  point  of  application 
is  given  by  equation  (3)  ;  the  respective  velocities  imparted  by 
the  force  being  shown  graphically  in  Fig.  27. 


EXAMPLES. 

i.  Given  Weight  of  rod  =  W  •=  100  Ibs., 
Length  of  rod  =  3  feet, 

Assume  g          =  32  feet  per  second, 

Force  applied  =  R  =       5  Ibs., 
Point  of  application  to  be  2.5  feet  from  one  end; 

determine  the  motion  imparted  to  the  rod  by  the  action  of  the  force  for 
one  second. 


48  APPLIED  MECHANICS. 

Solution. 
Equation  (i)  gives  us, 

5  =  ( 

Equation  (2)  gives, 

<2'5)(5)  -  (^p)  ©  (3)C/«  +  »/•),  or/  +  2/2  =  8 

.-.    /2  =  4.8,        /  =  -1.6. 

Hence  the  rod  at  the  end  nearest  the  force  acquires  a  velocity  of  4.8 
feet  per  second,  and  at  the  other  end  a  velocity  of  —1.6  feet  per 
second.  The  mean  velocity  is,  therefore,  1.6  feet  per  second;  and  we 
may  consider  the  rod  as  having  a  motion  of  translation  in  the  direc- 
tion of  the  force  with  a  velocity  of  1.6  feet  per  second,  and  a  rotation 
about  its  centre  with  such  a  speed  that  the  extreme  end  (i.e.,  a  point 
|  feet  from  the  centre)  moves  at  a  velocity  4.8  —  1.6  =  3.2  feet  per 

second.     Hence  angular  velocity  =  ^  =  2.14  per  second  =  122°. 6 

per  second.  , 

2.  Given  JF==  50  Ibs.,  /=  5  feet.     It  is  desired  to  impart  to  it, 
in  one  second,  a  velocity  of  translation  at  right  angles  to  its  length,  of  5 
feet  per  second,  together  with  a  rotation  of  4  turns  per  second :  find  the 
force  required,  and  its  point  of  application. 

3.  Assume  in  example    2  that   the  velocity  of  translation    is   in   a 
direction  inclined  45°  to  the  length  of  the  rod,  instead  of  90°.     Solve 
the  problem. 

4.  Given  a  force  of  3  Ibs.  acting  for  one-half  a  second  at  a  distance 
of  4  feet  from  one  end  of  the  rod,  and  inclined  at  30°  to  the  rod  : 
determine  its  motion. 

5.  Given  the  same  conditions  as  in  example  4,  and  also   a   force 
of  4  Ibs.,  parallel  and  opposite  in  direction  to  the  3-lb.  force,  and  acting 
also  for  one-half  a  second,  and  applied  at  3  feet  from  the  other  end : 
determine  the  resulting  motion. 


MOMENT  OF   THE   FORCES   CAUSING   ROTATION.  49 

6.  Given  two  equal  and  opposite  parallel  forces,  each  acting  at  right 
angles  to  the  length  of  the  rod,  and  each  equal  to  4  Ibs.,  one  being 
applied  at  i  foot  from  one  end,  and  the  other  at  the  middle  of  the  rod ; 
find  the  motion  imparted  to  the  rod  through  the  joint  action  of  these 
forces  for  one-third  of  a  second. 

§  48.  Moment  of  the  Forces  causing  Rotation.  —  Re- 
ferring again  to  Fig.  26,  and  considering  the  motion  of  the 
rod  as  a  combination  of  translation  and  rotation,  if  we  take 
moments  about  the  centre  C,  and  compare  the  total  moment 
of  the  forces  causing  the  rotation  alone,  whose  accelerations 
are  represented  by  the  triangles  aajbj),  with  the  total  moment 
of  the  actual  forces  acting,  whose  accelerations  are  represented 
by  the  trapezoid  AabB,  we  shall  find  these  moments  equal  to 
each  other  ;  for,  as  far  as  the  forces  represented  by  the  rectangle 
are  concerned,  every  elementary  force  nt(xy^dx  on  one  side  of 
the  centre  C  has  its  moment  (Cx)\m(xy^dx\  equal  and  opposite 
to  that  of  the  elementary  force  at  the  same  distance  on  the  other 
side  of  C :  hence  the  total  moment  of  the  forces  represented 
graphically  by  the  rectangle  AaJb^B  is  zero,  and  hence  — 

The  moment  about  C  of  those  represented  by  the  trapezoid 
•equals  the  moment  of  those  represented  by  the  triangles. 

Hence,  from  the  preceding,  and  from  what  has  been  pre- 
viously proved,  we  may  draw  the  following  conclusions  :  — 

i°.  If  a  force  be  applied  at  the  centre  of  the  rod,  it  will 
impart  the  same  velocity  to  each  particle. 

2°.  If  a  force  be  applied  at  a  point  different  from  the  centre, 
and  act  at  right  angles  to  its  length,  it  will  cause  a  translation 
of  the  rod,  together  with  a  rotation  about  the  centre  of  the  rod. 

3°.  In  this  latter  case,  the  moment  of  the  forces  imparting 
the  rotation  alone  is  equal  to  the  moment  of  the  single  resultant 
force  about  the  centre  of  the  rod,  and  the  velocity  of  translation 
imparted  in  a  unit  of  time  is  equal  to  the  number  of  units  of 
force  in  the  force,  divided  by  the  entire  mass  of  the  rod. 


APPLIED   MECHANICS. 


§49.  Effect  of  a  Pair  of  Equal  and  Opposite  Parallel 
Forces  applied  to  a  Straight  Rod  of  Uniform  Section  and 
Material.  —  Suppose  the  rod  to  be  AB  (Fig.  28),  and  let  the 
two  equal  and  opposite  parallel  forces  be  Dd  and  Ee,  each  equal 
to  F,  applied  at  D  and  E  respectively. 
The  mean  velocity  imparted  in  a  unit 

F 


of   time  by  either  force  will  be  —  ;  and, 

from  what  we  have  already  seen,  the  trap- 
ezoid  AabB  will  furnish  us  the  means  of 
representing  the  actual  velocity  imparted 
to  any  point  of  the  rod  by  the  force  Dd. 
The  relative  magnitudes  of  Aa  and  Bb,  the 
accelerations  at  the  ends,  will  depend,  of 

course,  on  the  position  of  D ;  but  we  shall 

77 

always    have    Cc  —  l(Aa  -f-  Bb)  =  — ,    a 

M 

quantity  depending  only  on  the  magnitude 
of  the  force.  So,  likewise,  the  trapezoid  AaJb^B  will  represent 
the  velocities  imparted  by  the  force  Ee ;  and  while  the  relative 
magnitude  of  Aal  and  Bbl  will  depend  upon  the  position  of  E, 

we  shall  always  have  Ccl  =  \(Aal  +  Bb,)  =  — .     Hence,  since 

Cc  =  Cc,,  the  centre  C  of  the  rod  has  no  motion  imparted  to  it 
by  the  given  pair  of  forces,  hence  the  motion  of  the  rod  is  one 
of  rotation  about  its  centre  C. 

The  resulting  velocity  of  any  point  of  the  rod  will  be  the 
difference  between  the  velocities  imparted  by  the  two  forces  ; 
and  if  these  be  laid  off  to  scale,  we  shall  have  the  second 
figure.  Hence  — 

A  pair  of  equal  and  opposite  parallel  forces,  applied  to  a 
straight  rod  of  uniform  section  and  material,  produce  a  rota- 
tion of  the  rod  about  its  centre.  Also,  — 

Such  a  rotation  about  the  centre  of  the  rod  cannot  be  pro- 


FIG.  28. 


EFFECT  OF  STATICAL    COUPLE    ON  STRAIGHT  ROD.       51 

duced  by  a  single  force,   but  requires  a  pair  of  equal  and  op- 
posite  parallel  forces. 

§  50.  Statical  Couple.  —  A  pair  of  equal  and  opposite 
parallel  forces  is  called  a  statical  couple. 

§51.  Effect  of  a  Single  Force  applied  at  the  Centre  of 
Gravity  of  a  Straight  Rod  of  Non-Uniform  Section  and 
Material.  —  In  the  case  of  a  straight  rod  of  non-uniform  sec- 
tion and  material,  we  may  consider  the  rod  as  composed  of  a 
set  of  particles  of  unequal  mass  :  and  if  we  imagine  each  par- 
ticle to  have  imparted  to  it  the  same  velocity  in  a  unit  of  time,, 
then,  using  the  same  method  of  graphical  representation  as 
before  (Fig.  24),  the  line  ab,  bounding  the  other  ends  of  the 
lines  representing  velocities,  will  be  parallel  to  AB ;  but  if  we 
were  to  represent  by  the  lines  xy,  not  the  velocities  imparted, 
but  the  forces  per  unit  of  length,  the  line  bounding  the  other 
ends  of  these  forces  would  not,  in  this  case,  be  parallel  to  AB. 
Moreover,  since  these  forces  are  proportional  to  the  masses,  and 
hence  to  the  weights  of  the  several  particles,  their  resultant 
would  act  at  the  centre  of  gravity  of  the  rod.  Hence  — 

A  force  applied  at  the  centre  of  gravity  of  a  straight  rod  will 
impart  the  same  velocity  to  each  point  of  the  rod ;  i.e.,  will  im- 
part to  it  a  motion  of  translation  only. 

§  52.  Effect  of  a  Statical  Couple  on  a  Straight  Rod  of 
Non-Uniform  Section  and  Material.  —  Let  such  a  rod  have 
imparted  to  it  only  a  motion  of  rotation  about  its  centre  of 
gravity,  and  let  us  adopt  the  same  modes  of  graphical  repre- 
sentation as  before. 

Let  the  origin  be  taken  at  O  (Fig.  29), 
the  centre  of  gravity  of  the  rod. 

Let  Aa   =  /,  =  velocity  imparted  to  A. 
Bb    =  /2  =  velocity  imparted  to  B. 
OA  =  a,  OB  =  b,  OC  =  x. 
CD  =  f  =  velocity  imparted  to  C. 
dM  =  elementary  mass  at  C. 


52  APPLIED   MECHANICS, 

Then,  from  similar  triangles,  we  have 

/_4*-4r, 

a  b 

and  hence  for  the  force  acting  on  dM  we  have 
dF=(CE)dx   =  ^xdM. 


Hence  the  whole  force  acting  on  AO,  and  represented  graph- 
ically by  Aa^Oy  is 

f     (*x---a 

J-  \    xdM, 
ajx  =  o 

and  that  acting  on  OB,  and  represented  by  B0bt,  is 

/     f*x  =  0  f     (*x  =  o 

-V2  /    xdM  =  J-  I    xdM. 
bjx  =  -b  ajx  =  -b 

Hence  for  the  resultant,  or  the  algebraic  sum,  of  the  two,  we 
have 


But  from  §  43  we  have  for  the  co-ordinate  x0  of  the  centre  of 
gravity  of  the  rod 


f, 


x  =  a 

xdM 


M 
but,  since  the  origin  is  at  the  centre  of  gravity,  we  have 

X0   =    O, 

and  hence 

\xdM  =  o        .-.     R  =  o. 

Jx=-6 

Hence  the  two  forces  represented  by  Aa,O  and  Bb,O  are  equal 
in  magnitude  and  opposite  in  direction  :  hence  the  rotation 
about  the  centre  of  gravity  is  produced  by  a  Statical  Couple. 


MEASURE   OF   THE  ROTATORY  EFFECT.  53 

Now,  a  train  of  reasoning  similar  to  that  adopted  in  the  case 
of  a  rod  of  uniform  section  and  material  will  show  that  a  single 
force  applied  at  some  point  which  is  not  the  centre  of  gravity 
of  the  rod  will  produce  a  motion  which  consists  of  two  parts  ;. 
viz.,  a  motion  of  translation,  where  all  points  of  the  rod  have 
equal  velocities,  and  a  motion  of  rotation  around  the  centre  of 
gravity  of  the  rod. 

§53.  Moment  of  a  Couple.  —  The  moment  of  a  statical 
couple  is  the  product  of  either  force  by  the  perpendicular  dis- 
tance between  the  two  forces,  this  perpendicular  distance  being 
called  the  arm  of  the  couple. 

§  54.  Measure  of  the  Rotatory  Effect.  —  Before  proceed- 
ing  to  examine  the  effect  of  a  statical  couple  upon  any  rigid 
body  whatever,  we  will  seek  a  means  of  measuring  its  effect  in 
the  cases  already  considered. 

The  measure  adopted  is  the  moment  of  the  couple ;  and,  in 
order  to  show  that  it  is  proper  to  adopt  this  measure,  it  will  be 
necessary  to  show  — 

That  the  moment  of  the  couple  is  proportional  to  the  angu- 
lar velocity  imparted  to  the  same  rod  in  a  unit  of  time ;  and 
from  this  it  will  follow  — 

That  two  couples  in  the  same  plane  with  equal  moments  will 
balance  each  other  if  one  is  right-handed  and  the  other  left-handed 

If  we  assume  the  origin  of  co-ordinates  at  C  (Fig.  30),  the 
centre  of  gravity  of  the  rod,  and  if  we 
denote  by  a  the  angular  velocity  imparted 
in  a  unit  of  time  by  the  forces  F  and  —  F, 
and  let  CD  —  *„  CE  =  XM  then  we  have 
for  the  linear  velocity  of  a  particle  situated 
at  a  distance  x  from  C  the  value 

CLT.  FIG.  30. 

The  force  which  will  impart  this  velocity  in  a  unit  of  time  to 

the  mass  dM  is 

axdM. 


54  APPLIED   MECHANICS. 

The  total  resultant  force  is 

afxtfM, 

which,  as  we  have  seen,  is  equal  to  zero.     The  moment  of  the 
elementary  force  about  C  is 


and  the  sum  of  the  moments  for  the  whole  rod  is 


and  this,  as  is  evident  if  we  take  moments  about  C,  is  equal  t* 
Fxi  -  Fx2  =  F(x,  -  x,)  =  F(DE). 

Now,  fxzdM  is  a  constant  for  the  same  rod  :  hence  any  quan- 
tity proportional  to  F(DE)  is  also  proportional  to  a. 

The  above  proves  the  proposition. 

Moreover,  we  have 

F(DE)  =  a 


whence  it  follows,  that  when  the  moment  of  the  couple  is  given, 
and  also  the  rod,  we  can  find  the  angular  velocity  imparted  in 
a  unit  of  time  by  dividing  the  former  by  fxzdM. 

§  55.   Effect   of  a  Couple  on  a  Straight  Rod  when  the 

Forces  are  inclined  to  the  Rod.  —  We  shall  next  show  that 

the  effect  of  such  a  couple  is  the  same  as  that  of  a  couple  of 

equal  moment  whose  forces  are  perpen- 

r%=^^.^  dicular  to  the  rod. 

/}         ^"-^^^  In  this  case  let  AD  and  BC  be  the 

forces  (Fig.  31).      The   moment    of   this 
couple  is  the  product  of  AD  by  the  per- 
c         *   pendicular  distance  between  AD  and  BC, 
the  graphical  representation  of  this  being 
the  area  of  the  parallelogram  ADBC. 


EFFECT  OF  A   STATICAL    COUPLE   ON  A   RIGID   BODY.    55 

Resolve  the  two  forces  into  components  along  and  at  right 
angles  to  the  rod.  The  former  have  no  effect  upon  the  motion 
of  the  rod  :  the  latter  are  the  only  ones  that  have  any  effect 
upon  its  motion.  The  moment  of  the  couple  which  they  form 
is  the  product  of  Ad  by  AB,  graphically  represented  by  paral- 
lelogram AdBb  ;  and  we  can  readily  show  that 

ADBC  =  AdBb. 

Hence  follows  the  proposition. 

§  56.   Effect  of  a  Statical  Couple  on  any  Rigid  Body.  — 

Refer  the  body  (Fig.  32)  to  three  rectan- 
gular axes,  OX,  OY,  and  OZ,  assuming 
the  origin  at  the  centre  of  gravity  of  the 
body,  and  OZ  as  the  axis  about  which 
the  body  is  rotating.  Let  the  mass  of  the 
particle  P  be  AJ/,  and  its  co-ordinates  be 


Then  will  the  force  that  would  impart 


FIG.  32. 


to  the  mass  AJf  the  angular  velocity  a  in  a  unit  of  time  be 


where  r  =.  perpendicular  from  P  on  OZ,  or 

r  —  ^x2  +  y2. 

This  force  may  be  resolved  into  two,  one  parallel  to  O  Y  an<f 
the  other  to  OX;  the  first  component  being  ax&M,  and  the 
second  a/y&M. 

Proceeding  in  the  same  way  with  each  particle,  and  finding 
the  resultant  of  each  of  these  two  sets  of  parallel  forces,  we 
shall  obtain,  finally,  a  single  force  parallel  to  OY  and  equal  to 


and  another  parallel  to  OX,  equal  to 


56  APPLIED   MECHANICS. 

But,  since  OZ  passes  through  the  centre  of  gravity  of  the  body, 
we  shall  have 

=  o     and     2A  M  =  o. 


Hence  the  resultant  is  in  each  case,  not  a  single  force,  but  a 
statical  couple.  Hence,  to  impart  to  a  body  a  rotation  about 
an  axis  passing  through  its  centre  of  gravity  requires  the  action 
of  a  statical  couple  ;  and  conversely,  a  statical  couple  so  applied 
will  cause  such  a  rotation  as  that  described. 

Further  discussion  of  the  motion  of  rigid  bodies  resulting 
from  the  action  of  statical  couples  is  unnecessary  for  our  pres- 
ent purpose,  hence  we  shall  pass  to  the  deduction  of  the  fol- 
lowing propositions,  viz.: 

PROP.  I.  Two  statical  couples  in  the  same  plane  balance 
each  other  when  they  have  equal  moments,  and  tend  to  pro- 
duce rotation  in  opposite  directions.  Let 
F1  at  a  and  —  Fl  at  b  represent  one 
couple  (left-handed  in  the  figure),  and 
let  Ft  at  d  and  —  F^  at  e  represent  the 
other  (right-handed  in  the  figure),  and  let 

F\ab]  =  FJ&)  ; 

then  will  these  two  couples  balance  each 
other. 

Proof.  —  The  resultant  of  Fl  at  a  and  F^ 
FlG>  33>  at  d  will  be  equal  in  amount,  and  directly 

opposed  to  the  resultant  of  —  Fl  at  b  and  —  Ft  at  e  and  both 
will  act  along  the  diagonal  fh  of  the  parallelogram  fchg. 
For  we  have  (fg)(ab)  =  (fc)(de)9  each  being  equal  to  the  area 
of  the  parallelogram. 

.    £«£•       .    £-.&. 

fg  "  f<  ^  ~f*  ' 

hence  follows  the  proposition. 

Hence  follows  that  for  a  couple  we  may  substitute  another 
in  the  same  plane,  having  the  same  moment,  and  tending  to 
rotate  the  body  in  the  same  direction. 


COUPLES   IN   THE   SAME   OR   PARALLEL   PLANES.        57 


FIG.  33  (a). 


PROP.  II.  Two  couples  in  parallel  planes  balance  each 
other  when  their  moments  are  equal,  and  the  directions  in 
which  they  tend  to  rotate  the  body  are  opposite. 

Let  (Fig.  33  (a))  the  planes  of  both  couples,  be  perpendicular 
to  OZ.  Reduce  them  both  so  as  to  have  their  arms  equal  and 
transfer  them,  each  in  its  own  plane, 
till  their  arms  are  in  the  X  plane. 
Let  ab  be  the  arm  of  one  couple, 
and  dc  that  of  the  other.  Then  will 
the  two  couples  form  an  equilibrate 
system.  For  the  resultant  of  the 
force  at  a  and  that  at  c  acts  at  e, 
and  is  twice  either  one  of  its  com- 
ponents, and  hence  is  equal  and  di- 
rectly opposed  to  the  resultant  of  the 
force  at  b  and  that  at  d. 

Hence  we  may  generalize  all  our  propositions  in  regard  to 
the  effect  of  statical  couples  and  we  may  conclude  that  — 

In  order  that  two  couples  may  have  the  same  effect,  it  is 
necessary  — 

i°.    That  they  be  in  the  same  or  parallel  planes. 

2°.    That  they  have  the  same  moment. 

3°.  That  they  tend  to  cause  rotation  in  the  same  direction 
(i.e.,  both  right-handed  or  both  left-handed  when  looked  at  from 
the  same  side}. 

It  also  follows,  that,  for  a  given  statical  couple,  we  may  sub- 
stitute another  having  the  magnitudes  of  its  forces  different, 
provided  only  the  moment  of  the  couple  remains  the  same. 

§57.  Composition  of  Couples  in  the  Same  or  Parallel 
Planes.  —  If  the  forces  of  the  couples  are  not 
the  same,  reduce  them  to  equivalent  couples 
having  the  same  force,  transfer  them  to  the 
same  plane,  and  turn  them  so  that  their  arms 
shall  lie  in  the  same  straight  line,  as  in  Fig. 
34;  the  first  couple  consisting  of  the  force  F 
at  A  and  —  F  at  B,  and  the  second  of  F  at  B  and  —F  at  C. 


L 


t 


FIG. 


APPLIED   MECHANICS. 


The  two  equal  and  opposite  forces  counterbalance  each  other, 
and  we  have  left  a  couple  with  force  F  and  arm 

AC  =  AB  +  PC 
.*.     Resultant  moment  =  F.  AC  =  F(AB)  +  F(BC). 

Hence  :  The  moment  of  the  couple  which  is  the  resultant  of 
two  or  more  couples  in  the  same  or  parallel  planes  is  equal  t~ 
the  algebraic  sum  of  the  moments  of  the  component  couples. 

EXAMPLES. 

i.  Convert  a  couple  whose  force  is  5  and  arm  6  to  an  equivalent 
couple  whose  arm  is  3.  Find  the  resultant  of  this  and  another  coupk 
in  the  same  plane  and  sense  whose  force  is  7  and  arm  8  ;  also  find  the 
force  of  the  resultant  couple  when  the  arm  is  taken  as  5. 


Solution. 


Moment  of  first  couple  =     5  x     6  =  30 

When  arm  is  3,  force  =     ^°-  =  10 

Moment  of  second  couple  =  7  x  8  =  56 
Moment  of  resultant  couple  =  30  -f  56  =  86 
When  arm  is  5,  force  =  -8^  =  17 

1.  Given  the  following  couples  in  one  plane  :  — 


Force. 

Arm. 

12 

17  1 

3 

8 

5 

7 

6 

9 

12 

12 

10 

9 

14 

6  J 

Force. 

5 


Arm. 


Convert  to  equivalent 
couples  having  the   < 
following :  — 


8 


20 


The  first  and  the  last  three  are  right-handed ;  the  second,  third,  and 
fourth  are  left-handed.  Find  the  moment  of  the  resultant  couple,  and 
also  its  force  when  it  has  an  arm  n. 


COUPLES  IN  PLANES  INCLINED    TO   EACH  OTHER.       59 


§58.  Representation  of  a  Couple  by  a  Line.  —  From  the 
preceding  we  see  that  the  effect  of  a  couple  remains  the  same 
as  long  as  — 

i  °.  Its  moment  does  not  change. 

2°.  The  direction  of  its  axis  (i.e.,  of  the  line  drawn  perpen- 
dicular to  tJie  plane  of  the  couple}  does  not  change. 

3°.  The  direction  in  which  it  tends  to  make  the  body  turn 
(right-handed  or  left-handed)  remains  the  same. 

Hence  a  couple  may  be  represented  by  drawing  a  line  in 
the  direction  of  its  axis  (perpendicular  to  its  plane),  and  laying 
off  on  this  line  a  distance  containing  as  many  units  of  length 
as  there  are  units  of  moment  in  the  couple,  and  indicating  by  a 
dot,  an  arrow-head,  or  some  other  means,  in  what  direction  one 
must  look  along  the  line  in  order  that  the  rotation  may  appear 
right-handed. 

This  line  is  called  the  Moment  Axis  of  the  couple. 

§  59.  Composition  of  Couples  situated  in  Planes  inclined 
to  Each  Other.  —  Suppose  we  have  two  couples  situated 
neither  in  the  same  plane  nor  in  parallel  planes,  and  that  we 
wish  to  find  their  resultant  couple.  We  may  proceed  as  fol- 
lows :  Substitute  for  them  equivalent  couples  with  equal  arms, 
then  transfer  them  in  their  own  plane  respectively  to  such  posi- 
tions that  their  arms  shall 
coincide,  and  lie  in  the 
line  of  intersection  of  the 
two  planes. 

This  having  been  done, 
let  OO,  (Fig.  35)  be  the 
common  arm,  F  and  —  F 
the  forces  of  one  couple, 
Fl  and  —  Ft  those  of  the 
other.  The  forces  F  and 
F,  have  for  their  resultant  R,  and  —  F  and  —F,  have  —R,. 
Moreover,  we  may  readily  show  that  R  and  —  R,  are  equal  and 


60  APPLIED   MECHANICS. 

parallel,  both  being  perpendicular  to  <9<92.  The  resultant  of 
the  two  couples  is,  therefore,  a  couple  whose  arm  is  OO^  and 
force  R,  the  diagonal  of  the  parallelogram  on  F  and  Flt  so  that 


R  =  \F2  -h  F*  +  2FFl  cos  0, 

where  0  is  the  angle  between  the  planes  of  the  couples.  Now, 
if  we  draw  from  O  the  line  Oa  perpendicular  to  OOI  and  to  F, 
and  hence  perpendicular  to  the  plane  of  the  first  couple,  and  if 
we  draw  in  the  same  manner  Ob  perpendicular  to  the  plane  of 
the  second  couple,  so  that  there  shall  be  in  Oa  as  many  units 
of  length  as  there  are  units  of  moment  in  the  first  couple,  and 
in  Ob  as  many  units  of  length  as  there  are  units  of  moment  in 
the  second  couple,  we  shall  have  — 

i°.  The  lines  Oa  and  Ob  are  the  moment  axes  of  the  two 
given  couples  respectively. 

2°.  The  lines  Oa  and  Ob  lie  in  the  same  plane  with  F  and 
FT,  this  plane  being  perpendicular  to  OOlt 

3°.  We  have  the  proportion 

Oa-.0b  =  F.  00,  -.F^OO^F:  F,. 


4°.  If  on  Oa  and  Ob  as  sides  we  construct  a  parallelogram, 
it  will  be  similar  to  the  parallelogram  on  F  and  /v  We  shall 
have  the  proportion 

Oc  :  R  =  Oa  :  F  =  Ob\F*\ 

and  since  the  sides  of  the  two  parallelograms  are  respectively 
perpendicular  to  each  other,  the  diagonals  are  perpendicular  to 
each  other  ;  and  since  we  have  also 

Oc  =  R  '•  Oa     and     Oa  =  F.  OO,        .'.     Oc  =  R  .  OOt, 
F 

it  follows  that  Oc  is  perpendicular  to  the  plane  of  the  resultant 
couple,  and  contains  as  many  units  of  length  as  there  are  units 
of  moment  in  the  moment  of  the  resultant  couple;  in  other 


COUPLE  AND  SINGLE  FORCE  IN  THE  SAME   PLANE.     6l 

words,  Oc  will   represent   the   moment  axis   of   the   resultant 
couple,  and  we  shall  have 


Oc  =  \Oa*  -f  Ob*  +  2Oa  . 
or,  if  we  let 

Oa  =  Z,     Ob  =  M,     Oc  =  G,     aOb  =  0, 


G  =  VZ2  +  J/2  +  2ZJ/cos6>. 

This  determines  the  moment  of  the  resultant  couple ;  and,  for 
the  direction  of  its  moment  axis,  we  have 


and 


sin  a  Oc  =  —  sin  6 
G 


—  —  sin0. 


Hence  we  can   compound  and  resolve  couples  just  as  we  do 
forces,  provided  we  represent  the  couples  by  their  moment  axes 

EXAMPLES. 

1.  Given     L  =  43,     M  —  15,     6  =  65°;  find  resultant  couple. 

2.  Given     Z  =  40,     M  =  30,     #  =  30° ;  find  resultant  couple. 

3.  Given     L  —     i,     M '  =     5,     #  =  45°;  find  resultant  couple. 

§  60.   Resultant  of  a  Couple  and  a  Single  Force  in  the 
Same  Plane.  —  Let  M  (Fig.  36  or  37)  be  the  moment  of  the 


wF 


FIG.  36.  FIG.  37. 

given  couple,  and  let  OF  =  F  be  the  single  force.  For  the 
given  couple  substitute  an  equivalent  couple,  one  of  whose 
forces  is  —  F  at  O,  equal  and  directly  opposed  to  the  single 


62 


APPLIED   MECHANICS, 


force  F,  these  two  counterbalancing  each  other,  and  leaving 
only  the  other  force  of  the  couple,  which  is  equal  and  parallel 
to  the  original  single  force  F,  and  acts  along  a  line  whose 


M 

distance  from  O  is  OA  =  — . 

F 


Hence  — 


The  resultant  of  a  single  force  and  a  couple  in  the  same  plane 
is  a  force  equal  and  parallel  to  the  original  force,  having  its 
line  of  direction  at  a  perpendicular  distance  from  the  original 
force  equal  to  the  moment  of  the  couple  divided  by  the  force. 

Fig.  36  shows  the  case  when  the  couple  is  right-handed,  and 
Fig.  37  when  it  is  left-handed. 

§61.  Composition  of  Parallel  Forces  in  General.  —  In 
each  case  of  composition  of  parallel  forces  (§§  34,  35,  and  36) 
it  was  stated  that  the  method  pursued  was  applicable  to  all 
cases  except  those  where 

^F=  o. 

We  were  obliged,  at  that  time,  to  reserve  this  case,  because  we 
had  not  studied  the  action  of  a  statical  couple ;  but  now  we  will 
ad  pt  a  method  for  the  composition  of  parallel  forces  which  will 
apply  in  all  cases. 

(a)  When  all  the  forces  are  in  one  plane.  Assume,  as  we  did 
in  §35,  the  axis  OY  to  be  parallel  to 
the  forces ;  assume  the  forces  and  the 
co-ordinates  of  their  lines  of  direction, 
as  shown  in  the  figure  (Fig.  38).  Now 
place  at  the  origin  O,  along  OY,  two 
equal  and  opposite  forces,  each  equal  to 
x  F, ;  then  these  three  forces,  viz.,  F,  at 
D,  OA,  and  OB,  produce  the  same  effect 
as  F,  at  D  alone ;  but  F,  at  D  and  OR 
form  a  couple  (left-handed  in  the  figure) 
whose  moment  is  —F,*,.  Hence  the 
force  Ft  is  equivalent  to  — 


COMPOSITION  OF  PARALLEL   FORCES. 


i°.  An  equal  and  parallel  force  at  the  origin,  and 

2°.  A  statical  couple  whose  moment  is  —P\x^. 

Likewise  the  force  F2  is  equivalent  to  (i°)  an  equal  and  par- 
allel force  at  the  origin,  and  (2°)  a  couple  whose  moment  is 
-F2x2,  etc. 

Hence  we  shall  have,  if  we  proceed  in  the  same  way  with 
all  the  forces,  for  resultant  of  the  entire  system  a  single  force 

R  =  ^F  along  OY, 
and  a  single  resultant  couple 


(Observe  that  downward  forces  and  left-handed  couples  are 
to  be  accounted  negative.) 

Now,  there  may  arise  two  cases. 
i°.  When  ^F—  o,  and 
2°.  When  2F><0. 

CASE  I.  When  ^F  =  o,  the  resultant  force  along  0  Y  van- 
ishes, and  the  resultant  of  the  entire  system  is 
a  statical  couple  whose  moment  is 


CASE  II.  When  %F  >  <  o,  we  can  reduce 
the  resultant  to  a  single  force. 

Let  (Fig.  39)  OB  represent  the  resultant 
force  along  OY,  R  =  %F.  With  this,  compound 
the  couple  whose  moment  is  M  —  —  2<Fx,  and 
we  obtain  as  resultant  (§  60)  a  single  force 


FIG.  39. 


whose  line  of  action  is  at  a  perpendicular  distance  from  OY 
equal  to 

AO    =    Xr   = 


64 


APPLIED  MECHANICS. 


(b)  When  the  forces  are  not  confined  to  one  plane.  Assume, 
as  before  (Fig.  40),  OZ  parallel  to  the  forces,  and  let  F  acting 

through  A  be  one  of  the  given 
forces,  the  co-ordinates  of  A  be- 
ing x  and  y.  Place  at  O  two  equal 
and  opposite  forces,  each  equal  to 
F,  and  also  at  B  two  equal  and 
opposite  forces,  each  equal  to  F. 
These  five  forces  produce  the 
same  effect  as  F  alone  at  A,  and 
they  may  be  considered  to  con- 
sist of  — 
i°.  A  single  force  F  at  the  origin. 

2°.  A  couple  whose  'forces  are  F  at  B  and  —  F  at  O,  and 
whose  moment  is  —  Fx  acting  in  the  y  plane. 

3°.  A  couple  whose  forces  are  F  at  A  and  —  F  at  B,  and 
whose  moment  is  Fy  acting  in  the  x  plane.  Treating  each  of 
the  forces  in  the  same  way,  we  shall  have,  in  place  of  the  entire 
system  of  parallel  forces,  the  following  forces  and  couples  :  — 


FIG.  40. 


i°.  A  single  force  R 
2°.  A  couple  My  = 
3°.  A  couple  Mx  = 
Now,  there  may  be 
two  cases  :  — 

i°.  When  2F  ><  O. 
2°.  When  $F  =  o. 


along  OZ. 
in  the  y  plane. 
in  the  x  plane. 


CASE  I.  When 
<  o,  we  can  reduce  to  a 
single  resultant  force 
having  a  fixed  line  of 
direction.  Lay  off  (Fig. 
4 1 )  along  OZ,  OH  —  $F.  FIG.  4X. 

Combining  this  with  the  first  of  the  above-stated  couples,  we 


R--SF 


COMPOSITION  OF  PARALLEL   FOKCE.1. 


obtain  a  force  R  =  2<F  at  A,  where  OA  =  -'• —  —  xr.      Then 

2F 

combine  with  this  resultant  force  R  —  2F  at  A,  the  second 
couple,  and  we  shall  have  as  single  resultant  of  the  entire 
system  a  single  force 

R  =  2F 

acting  through  B,  where 


Hence  the  resultant  is  a  force  whose  magnitude  is 

R  —  2/7, 

the  co-ordinates  of  its  line  of  direction  being 

CASE  II.  When  S/7  —  o,  there  is  no  single  resultant  force  ; 
but  the  system  reduces  to  two  couples,  one  in  the  x  plane  and 
one  in  the  y  plane,  and  these  two  can  be  reduced  to  one  single 
resultant  couple.  (Observe  that  couples  are  to  be  accounted 
positive  when,  on  being  looked  at  by  the  observer  from  the  posi- 
tive part  of  the  axis  towards 
the  origin,  they  are  t  right- 
handed  ;  otherwise  they  are 
negative.) 

The  moment  axis  of  the 
couple  in  the  x  plane  will 
be  laid  off  on  the  axis  OX 
from  the  origin  towards  the 
positive  side  if  the  moment 
is  positive,  or  towards  the 
negative  side  if  it  is  nega-  FIG.  42. 

tive,  and  likewise  for  the  couple  in  the  y  plane. 


— -~ x 


66 


APPLIED   MECHANICS. 


Hence  lay  off  (Fig.  42)  OB  —  Mx,  OA  =  My)  and  by 
completing  the  rectangle  we  shall  have  OD  as  the  moment 
axis  of  the  resultant  couple  ;  hence  the  resultant  couple  lies 
in  a  plane  perpendicular  to  OD,  and  its  moment  bears  to 
OD  the  same  ratio  as  Mx  bears  to  OB. 

Hence  we  may  write 

OD  =  Mr  = 


cosDOX    =         =  cos0. 
Mr 

If  My  had  been  negative,  we  should  have  OR  as  the  moment 
axis  of  the  resultant  couple. 


EXAMPLES. 


p. 

X. 

y- 

P. 

X. 

>'• 

I. 

5 

4 

3 

2. 

5 

-4 

3 

3 

2 

i 

—  2 

2 

—  i 

i 

3 

5 

-3 

3 

5 

Find  the  resultant  in  each  example. 

§62.  Resultant  of  any  System  of  Forces  acting  at  Dif- 

ferent Points  of  a  Rigid 
Body,  all  situated  in  One 
Plane.  —  Let  CF  =  F  (Fig. 
43)  be  one  of  the  given  forces. 
Let  all  the  forces  be  referred 
to  a  system  of  rectangular 
axes,  as  in  the  figure,  and  let 
a  =  angle  made  by  F  with 
FIG.  43.  OX,  etc.  Let  the  co-ordi- 


M   A 


nates  of  the  point  of  application  of  F  be  AO  =  x,  BO  = 


SYSTEM  OF  FORCES  ACTING    ON  RIGID   BODY.  6? 

We  first  decompose  CF  =  F  into  two  components,  parallel 
respectively  to  OX  and  O  Y.  These  components  are 

CD  =  Fcosa,     CE  =  Fsina. 

Apply  at  O  in  the  line  O  Y  two  equal  and  opposite  forces,  each 
equal  to  Fsin  a,  and  at  O  in  the  line  OX  two  equal  and  opposite 
forces,  each  equal  to  Fcosa.  Since  these  four  are  mutually 
balanced,  they  do  not  alter  the  effect  of  the  single  force ;  and 
hence  we  have,  in  place  of  Fat  C,  the  six  forces  CD,  OM,  OK, 
CE,  ON,  OG.  Of  these  six,  CE  and  OG  form  a  couple  whose 
moment  is 

—  (Fsma)x  =  —Fxsina, 

CD  and  OK  form  a  couple  whose  moment  is 
(Fcosa)y  =  Fycosa. 

These  two  couples,  being  in  the  same  plane,  give  as  result- 
ant moment  their  algebraic  sum,  or 

F(y  cos  a  —  x  sin  a) . 

We  have,  therefore,  instead  of  the  single  force  at  C,  the  follow- 
ing:— 

i°.   OM  —  Fcos  a  along  OX. 

2°.    ON  =  Fsin  a  along  O  Y. 

3°.  The  couple  M  —  F(y  cos  a  —  x  sin  a)  in  the  given  plane. 

Decompose  in  the  same  way  each  of  the  given  forces ;  and 
we  have,  on  uniting  the  components  along  OX,  those  along  OY, 
and  the  statical  couples  respectively,  the  following:  — 

i°.  A  resultant  force  along  OX,  Rx  —  ^Fcos  a. 

2°.  A  resultant  force  along  OY,  Ry  —  ^Fsma. 

3°.  A  resultant  couple  in  the  plane,  whose  moment  is 

M  =  %FLi>  cos  a  —  x  sin  a}. 


68 


APPLIED   MECHANICS. 


This    entire    system,   on    compounding   the   two  forces   at   Ot 
reduces  to 


making  with  OX  an  angle  ar,  where 

^F  cos  a 


cos  OT  = 


R 


2°.  A  resultant  couple  in  the  same  plane,  whose  moment  is 
M  =  *%F(y  cos  a  —  x  sin  a)  . 


Now  compound  this  resultant  force  and  couple,  and  we  have, 
Y  for  final  resultant,  a  single 

force  equal  and  parallel  to 
BE     R,  and  acting  along  a  line 
whose    perpendicular    dis- 
tance from  O  is  equal  to 

M 

R' 


G 

-\ 

,-- 

A*                 R 

C                  I 

^^ 

L                 0 

E 

H 

FIG.  44. 

Suppose  (Fig.  44)  the  force 

OB  =  ^F  cos  a, 
614  = 
OR  = 


The  equation  of   this   line 
may  be  found  as  follows  : 


+  (S^1  sin  a)2; 


and  let  us  suppose  the  resultant  couple  to  be  right-handed,  and 
let 


then  will  the  line  ME  parallel  to  OR  be  the  line  of  direction 
•of  the  single  resultant  force. 


CONDITIONS   OF  EQUILIBRIUM.  69 

Assuming  the  force  R  to  act  at  any  point  C  (xr,  yr)  of  this 
line,  if  we  decompose  it  in  the  same  way  as  we  did  the  single 
forces  previously,  we  obtain  — 

i°.  The  force  R  cos  ar  =  2^  cos  a  along  OX. 

2°.  The  force  R  sinar  =  XFsina  along  OY. 

3°.  The  couple  R(yr  cos  ar  —  xr  sin  a^). 

Hence  we  must  have 

R(yr  cos  ar  —  xr  sin  ar)  =  ^F(y  cos  a  —  x  sin  a)  =  J/~. 

Hence  for  the  equation  of  the  line  of  direction  we  have 

M 

yr  cos  ar  —  xr  sin  ar  =  —  .  (  i  j 

R 

Another  form  for  the  same  equation  is 

cos  a)  _  Xr(2J?sma)  =  M.  (2) 


§  63.  Conditions  of  Equilibrium.  —  If  such  a  set  of  forces 
be  in  equilibrium,  there  must  evidently  be  no  tendency  to  h-an<r 
lation  and  none  to  rotation.  Hence  we  must  have 

R  =  o     and     M  =  o. 

Hence  the  conditions  of  equilibrium  for  any  system  of  force? 
in  a  plane  are  three  ;  viz.,  — 

2/7  cos  a  =  o,     2/7  sin  a  =  o,     2/7(j>cosa  —  ^sina)  =  o. 

Another  and  a  very  convenient  way  to  state  the  conditions  of 
equilibrium  for  this  case  is  as  follows  :  — 

If  the  forces  be  resolved  into  components  along  two  direction? 
at  right  angles  to  each  other,  then  the  algebraic  sum  of  the  com- 
ponents along  each  of  these  directions  must  be  zero,  and  th* 
algebraic  sum  of  the  moments  of  the  forces  about  any  axis 
pendicular  to  the  plane  of  the  forces  must  equal  zero. 


APPLIED   MECHANICS. 


EXAMPLES. 


i.  Given 


2.  Given 


p. 

X. 

y> 

5 

3 

2 

10 

i 

3 

-7 

4 

2 

P. 

X. 

* 

12 

27 

3 

-5       - 


54 


30° 

45° 


Find  the  resultant,  and 
the  equation  of  its 
line  of  direction. 

Find  the  resultant,  and 
the  equation  of  its 
line  of  direction. 


§64.  Resultant  of  any  System  of   Forces   not  confined 

to  One  Plane Suppose  we 

have  a  number  of  forces  applied 
at  different  points  of  a  rigid 
body,  and  acting  in  different 
directions,  of  which  we  wish  to 
find  the  resultant.  Refer  them 
all  to  a  system  of  three  rect- 
x  angular  axes,  OX,  OY,  OZ 
(Fig.  45).  Let  PR  =  F  be 
one  of  the  given  forces.  Re- 
solve it  into  three  components, 
PK,  PH,  and  PG,  parallel 
Let 


FIG.  45- 

respectively  to  the  three  axes. 


RPK  =  a,     RPH  = 


RPG 


Let  OA  —  x,  OB  =  y,  OC  —  z,  be  the  co-ordinates  of  the 
point  of  application  of  the  force  F.  Now  introduce  at  B  and 
also  at  O  two  forces,  opposite  in  direction,  and  each  equal  to  PK. 
We  now  have,  instead  of  the  force  PK,  the  five  forces  PK,  BM, 
BN,  OS,  and  OT.  The  two  forces  PK  and  BN  form  a  couple 
in  the  y  plane,  whose  axis  is  a  line  parallel  to  the  axis  OY,  and 
whose  moment  is  (PK)(EB)  —  (Fcosa)z  =  Fzcosa.  The 


FORCES  NOT  CONFINED    TO   ONE   PLANE.  fl 

forces  £Mand  OT  form  a  couple  in  the  z  plane,  whose  moment 

is 

(BM)(OB}  =  -Jycosa. 

Now  do  the  same  for  the  other  forces  PH  and  PG,  and  we  shall 
finally  have,  instead  of  the  force  PR,  three  forces, 

F  cos  a,     F  cos  ft     F  cos  y, 

acting  at  O  in  the  directions  OX,  O  V,  and  OZ  respectively, 
together  with  six  couples,  two  of  which  are  in  the  x  plane,  two 
in  the  y  plane,  and  two  in  the  z  plane. 

They  thus  form  three  couples,  whose  moments  are  as  fol- 

lows :  — 

Around  OX,  F(y  cos  y  —  z  cos  /?)  ; 
Around  OY,  F(zcosa  —  #cosy); 
Around  OZ,  F(x  cos  ft—y  cos  a)  . 

Treat  each  of  the  given  forces  in  the  same  way,  and  we  shall 
have,  in  place  of  all  the  forces  of  the  system,  three  forces, 

^F  cos  a  along  OX, 
^F  cos  J3  along  OY, 
along  OZ; 


and  three  couples,  whose  moments  are  as  follows  :  — 

Around  OX,  Mx  —  ^F(y  cos  y  —  z  cos  ft)  ; 
Around  O  Y,  My  =  ^F(z  cos  a  —  x  cos  y)  ; 
Around  OZ,  Mz  —  2F(xcos(3  —  jycosa). 


The  three  forces  give  a  resultant  at  O  equal  to 


R  =  V(cosa)2  -f  (XFcos/3)2  4-  -&F  cosy)2,  (i) 


a  0  (   . 

cosar  =  -  -  —  ,      cos  ft-  =  -  ~—  S     cosyr  =  -  -  —  *-.    (2) 

.  K 


APPLIED   MECHANICS. 


For  the  three  couples  we  have  as  resultant 


-»-* 


COS /a  = 


M' 


COS  v  = 


Mz 


(3) 

(4) 


A,  p,  and  v  being  the  angles  made  by  the  moment  axis  of  the 
resultant  couple  with  OX,  O  Y,  and  OZ  respectively. 

Thus  far  we  have  reduced  the  whole  system  to  a  single  result- 
ant force  at  the  origin,  and  a  couple.  Sometimes  we  can  reduce 

the  system  still  farther, 
and  sometimes  not.  The 
following  investigation  will 
show  when  we  can  do  so. 
Let  (Fig.  46)  OP  —  R  be 
the  resultant  force,  and 
OC  =M  the  moment  axis 
of  the  resultant  couple. 
Denote  the  angle  between 
them  by  6  (a  quantity  thus 
far  undetermined).  Pro- 
ject OP  =  R  on  OC.  Its 
projection  will  be  OD  =  RcosO;  then  project,  in  its  stead,  the 
broken  line  OABP  on  OC.  By  the  principles  of  projections, 
the  projection  of  this  broken  line  will  equal  OD. 

Now  OA,  AB,  and  BP  are  the  co-ordinates  of  P,  and  make 
with  OC  the  same  angle  as  the  axes  OX,  OY,  and  OZ ;  i.e., 
A.,  //,,  and  v  respectively :  hence  the  length  of  the  projection  is 


FIG.  46. 


But 


Hence 


OA  = 


OAcosX  + 

AB  = 


R  COS  0  =   R  COS  Or  COS  A. 

COS0         =    COS  Or  COS  A.         +   COS  fir  COS  fJL 


BP  =  £cosyr. 
R  cos  pr  cos  p,  -f  ^cosyrcosv 


+  cos  yr  cosv.       (5) 


CONDITIONS   OF  EQUILIBRIUM.  73 

This  enables  us  to  find  the  angle  between  the  resultant  force 
and  the  moment  axis  of  the  resultant  couple. 

The  following  cases  may  arise  :  — 

i°.  When  cos  0  —  o,  or  6  —  90°,  the  force  lies  in  the  plane 
of  the  couple,  and  we  can  reduce  to  a  single  force  acting  at  a 

distance  from   O  equal  to  —  ,  and  parallel  to  R  at  O. 

R 

2°.  When  cos  0  =  I,  or  0  —  o,  the  moment  axis  of  the 
couple  coincides  in  direction  with  the  force  :  hence  the  plane 
of  the  couple  is  perpendicular  to  the  force,  and  no  farther 
reduction  is  possible. 

3°.  When  0  is  neither  o°  nor  90°,  we  can  resolve  the  couple 
M  into  two  component  couples,  one  of  which,  McosO,  acts  in  a 
plane  perpendicular  to  the  direction  of  R,  and  the  other,  J/sin  0, 
acts  in  a  plane  containing  R.  The  latter,  on  being  combined 
with  the  force  R  at  the  origin,  gives  an  equal  and  parallel  force 
whose  line  of  action  is  at  a  distance  from  that  of  R  at  O,  equal 
to 

MsmO 
R 

4°.  When  M  =  o,  the  resultant  is  a  single  force  at  O. 

5°.  When  R  —  o,  the  resultant  is  a  couple. 

§65.  Conditions  of  Equilibrium.  —  To  produce  equilibrium, 
we  must  have  no  tendency  to  translation  and  none  to  rotation. 
Hence  we  must  have 

R  =  o     and    M  =  o. 
Hence  we  have,  in  general,  six  conditions  of  equilibrium  ;  viz.,  — 


a  =  o,     2,J?cos/3  ==  o,     ^F  cos  7  ==  o. 
=  o,    My          =  o,    Mz          =  a 


74  APPLIED  MECHANICS. 


EXAMPLES. 

1.  Prove  that,  whenever  three  forces  balance  each  other,  they  must 
lie  in  one  plane. 

2.  Show  how  to  resolve  a  given  force  into  two  whose  sum  is  given, 
the  direction  of  one  being  also  given. 

3.  A  straight  rod  of  uniform  section  and  material  is  suspended  by  two 
strings  attached  to  its  ends,  the  strings  being  of  given  length,  and  attached 
to  the  same  fixed  point :  find  the  position  of  equilibrium  of  the  rod. 

4.  Two  spheres  are  supported  by  strings  attached  to  a  given  point, 
and  rest  against  each  other :  find  the  tensions  of  the  strings. 

5 .  A  straight  rod  of  uniform  section  and  material  has  its  ends  resting 
against  two  inclined  planes  at  right  angles  to  each  other,  the  vertical 
plane  which  passes  through  the  rod  being  at  right  angles  to  the  line  of 
intersection  of  the  two  planes  :  find  the  position  of  equilibrium  of  the 
rod,  and  the  pressure  on  each  plane,  disregarding  friction. 

6.  A  certain  body  weighs  8  Ibs.  when  placed  in  one  pan  of  a  false 
balance  of  equal  arms,  and  10  Ibs.  in  the  other :  find  the  true  weight  of 
the  body. 

7.  The  points  of  attachment  of  the  three  legs  of  a  three-legged  table 
are  the  vertices  of  an  isosceles  right-angled  triangle  ;  a  weight  of  100  Ibs. 
is  supported  at  the  middle  of  a  line  joining  the  vertex  of  one  of  the  acute 
angles  with  the  middle  of  the  opposite  side :    find  the  pressure  upon 
each  leg. 

8.  A  heavy  body  rests  upon  an  inclined  plane  without  friction  :  find 
the  horizontal  force  necessary  to  apply,  to  prevent  it  from  falling. 

9.  A  rectangular  picture  is  supported   by  a  string   passing  over  a 
smooth  peg,  the  string  being  attached  in  the  usual  way  at  the  sides,  but 
one-fourth  the  distance  from  the  top  :  find  how  many  and  what  are  the 
positions  of  equilibrium,  assuming  the  absence  of  friction. 

16.  Two  equal  and  weightless  rods  are  jointed  together,  and  form  a 
right  angle ;  they  move  freely  about  their  common  point :  find  the 
ratio  of  the  weights  that  must  be  suspended  from  their  extremities,  that 
one  of  them  may  be  inclined  to  the  horizon  at  sixty  degrees. 

ii.  A  weight  of  100  Ibs.  is  suspended  by  two  flexible  strings,  one 
of  which  is  horizontal,  and  the  other  is  inclined  at  an  angle  of  thirty 
degrees  to  the  vertical :  find  the  tension  in  each  string. 


D  YNAMICS.  —  DEFINITIONS.  ?$ 


CHAPTER  II. 

DYNAMICS. 

§  66.  Definitions  --  Dynamics  is  that  part  of  mechanics 
which  discusses  the  forces  acting,  when  motion  is  the  result. 

Velocity,  in  the  case  of  uniform  motion,  is  the  space  passed 
over  by  the  moving  body  in  a  unit  of  time  ;  so  that,  if  s  repre- 
sent the  space  passed  over  in  time  tt  and  v  represent  the  velocity, 
then 


Velocity,  in  variable  motion,  is  the  limit  of  the  ratio  of  the 
space  (AJ-)  passed  over  in  a  short  time  (A/),  to  the  time,  as  the 
latter  approaches  zero  :  hence 

r-* 

dt 

Acceleration  is  the  limit  of  the  ratio  of  the  velocity  ^A  ;  Im- 
parted  to  the  moving  body  in  a  short  time  (A/),  to  the  time,  as 
the  time  approaches  zero.  Hence,  if  a  represent  the  accelera- 
tion, 


*• 


76  APPLIED   MECHANICS. 

§  67.   Uniform  Motion In  this  case  the  acceleration  is 

zero,  and  the  velocity  is  constant ;  and  we  have  the  equation 


s  =  vt. 


§  68.   Uniformly  Varying  Motion.  —  In  this  case  the  ac- 
celeration is  constant :  hence  a  is  a  constant  in  the  equation 


and  we  obtain  by  one  integration 

ds 
v  =  -  =  „+,,. 

where  c  is  an  arbitrary  constant  :  to  determine  it  we  observe, 
that,  if  v0  represent  the  value  of  v  when  /  =  o,  we  shall  have 

v0  =  o  -f  c 
.'.    c    =  v0 


and  by  another  integration 

s  = 

-.vJiera  s  is  the  space  passed  over  in  time  //  the  arbitrary  con- 
sfant  vanishing,  because,  when  /  =  o,  s  is  also  zero. 

§  69.  Measure  of  Force.  —  It  has  already  been  seen,  that, 
when  a  body  is  either  at  rest  or  moving  uniformly  in  a  straight 
line,  there  are  either  no  forces  acting  upon  it,  or  else  the  forces 
actr  •>  upon  it  are  balanced.  If,  on  the  other  hand,  the  motion 
of  -<>e  body  is  rectilinear,  but  not  uniform,  the  only  unbalanced 
force  acting  is  in  the  direction  of  the  motion,  and  equal  in  mag- 
nitude to  the  momentum  imparted  in  a  unit  of  time  in  the  direc- 
tion of  the  motion,  or,  in  other  words,  to  the  limit  of  the  ratio 
of  the  momentum  imparted  in  a  short  time  (A*),  to  the  time,  as 
the  latter  approaches  zero. 


MECHANICAL    WORK.  —  UNIT  OF   WORK.  // 

Thus,  if  F  denote  the  force  acting  in  the  direction   of  the 
motion,  m  the  mass,  and  a  the  acceleration,  we  shall  have 

,.,  dv  d2s  (i) 

F  =  ma  =  m  —  =  m  — .  v  ' 

dt  dt2 

From  (i)  we  derive 

mdv  =  Fdt;  (2) 

and,  if  VQ  be  the  velocity  of  the  moving  body  at  the  time  when 
/  =  4,  and  2/x  its  velocity  when  /  =  tlt  we  shall  have 


Xvi  r> 

mdv  =  I 
Jto 


Fdt 

JVQ  J  tQ 

or 

m(v,  -  v0)  =  J   *Fdt;  (3) 

or,  in  words,  the  momentum  imparted  to  the  body  during  the 
time  /  =  (/,  —  /0)  by  the  force  F,  will  be  found  by  integrating 
the  quantity  Fdt  between  the  limits  /x  and  tQ. 

§  70.  Mechanical  Work.  —  Whenever  a  force  is  applied  to 
a  moving  body,  the  force  is  either  used  in  overcoming  resist- 
ances (i.e.,  opposing  forces,  such  as  gravity  or  friction),  and 
leaving  the  body  free  to  continue  its  original  motion  undis- 
turbed, or  else  it  has  its  effect  in  altering  the  velocity  of  the 
body.  In  either  case,  the  work  done  by  the  force  is  the  prod- 
uct of  the  force,  by  the  space  passed  through  by  the  body  *n 
the  direction  of  the  force. 

Unit  qf  Work.  —  The  unit  of  work  is  that  work  which  is 
done  when  a  unit  of  force  acts  through  a  unit  of  distance  in 
the  same  direction  as  the  force ;  thus,  if  one  pound  and  one 
foot  are  our  units  of  force  and  length  respectively,  the  unit  of 
work  will  be  one  foot-pound. 

If  a  constant  force  act  upon  a  moving  body  in  the  direction 
of  its  motion  while  the  body  moves  through  the  space  s,  the 
work  done  by  the  force  is 

Fs; 


APPLIED  MECHANICS. 


and  this,  if  the  force  is  unresisted,  is  the  energy,  or  capacity  for 
performing  work,  which  is  imparted  to  the  body  upon  which  the 
force  acts  while  it  moves  through  the  space  s. 

Thus,  if  a  lo-pound  weight  fall  freely  through  a  height  of 
5  feet,  the  energy  imparted  to  it  by  the  force  of  gravity  during 
this  fall  is  10  X  5  =  50  foot-pounds,  and  it  would  be  necessary 
to  do  upon  it  50  foot-pounds  of  work  in  order  to  destroy  the 
velocity  acquired  by  it  during  its  fall.  If,  on  the  other  hand, 
the  force  is  a  variable,  the  amount  of  work  done  in  passing 
over  any  finite  space  in  its  own  direction  will  be  found  by  in- 
tegrating, between  the  proper  limits,  the  expression 


The  power  which  a  machine  exerts  is  the  work  which  it 
performs  in  a  unit  of  time. 

The  unit  of  power  commonly  employed  is  the  horse-power, 
which  in  English  units  is  equal  to  33000  foot-pounds  per 
minute,  or  550  foot-pounds  per  second. 

§71.  Energy.  —  The  energy  of  a  body  is  its  capacity  for 
performing  work. 

Kinetic  or  Actual  Energy  is  the  energy  which  a  body  pos- 
sesses in  virtue  of  its  velocity  ;  in  other  words,  it  is  the  work 
necessary  to  be  done  upon  the  body  in  order  to  destroy  its 
velocity.  This  is  equal  to  the  work  which  would  have  to  be 
done  to  bring  the  body  from  a  state  of  rest  to  the  velocity  with 
which  it  is  moving.  Assume  a  body  whose  mass  is  m,  and  sup- 
pose that  its  velocity  has  been  changed  from  VQ  to  vv  Then  if 
F  be  the  force  acting  in  the  direction  of  the  motion,  we  shall 
have,  from  equation  (2),  §  69,  that 

Fvdt  =  mvdv;  (i) 

but 

vdt     =  ds 

/.    Fds    =  mvdv.  (2) 


ATWOOD'S  MACHINE.  79 

Hence,  by  integration, 

I  mvdv  =    /  Fds 

*Jvo  *J 

/.     \m(v*  -  V02)  =  fFds;  (3) 

but  fFds  is  the  work  that  has  been  done  on  the  body  by  the 
force,  and  the  result  of  doing  this  work  has  been  to  increase 
its  velocity  from  v0  to  vt.  It  follows,  that,  in  order  to  change 
the  velocity  from  v0  to  vu  the  amount  of  work  necessary  to  per- 
form upon  the  body  is 

*«(*,*  -  *>o2)  =  i  —  (z>x*  -  *>o2).  (4) 

6 

If  v0  =  o,  this  expression  becomes 

\mv*,  or  ^  (5) 

2g 

which  is  the  expression  for  the  kinetic  energy  of  a  body  of  mass 
m  moving  with  a  velocity  vt. 

§  72.  Atwood's  Machine.  —  A  particular  case  of  uniformly 
accelerated  motion  is  to  be  found  in  Atwood's  machine,  in  which 
a  cord  is  passed  over  a  pulley,  and  is  loaded  with  unequal  weights 
on  the  two  sides.  Were  the  weights  equal,  there  would  be  no 
unbalanced  force  acting,  and  no  motion  would  ensue ;  but  when 
they  are  unequal,  we  obtain  as  a  result  a  uniformly  accelerated 
motion  (if  we  disregard  the  action  of  the  pulley),  because  we 
have  a  constant  force  equal  to  the  difference  of  the  two  weights 
acting  on  a  mass  whose  weight  is  the  sum  of  the  two  weights. 
Thus,  if  we  have  a  lo-pound  weight  on  one  side  and  a  5-pound 
weight  on  the  other,  the  unbalanced  force  acting  is 

F  =  io-  —  5  =  5  Ibs. 


SO  APPLIED   MECHANICS. 


T  O    "  i_     f 

The  mass  moved  is  M  ==  -  3UL  :  hence  the  resulting  ac- 

£• 
celeration  is 


§  73.  Normal  and  Tangential  Components  of  the  Forces 
acting  on  a  Heavy  Particle.  —  If  a  body  be  in  motion,  either 
in  a  straight  or  in  a  curved  line,  and  if  at  a  certain  instant  all 
forces  cease  acting  on  it,  the  body  will  continue  to  move  at  a 
uniform  rate  in  a  straight  line  tangent  to  its  path  at  that  point 
where  the  body  was  situated  when  the  forces  ceased  acting. 

If  an  unresisted  force  be  applied  in  the  direction  of  the 
body's  motion,  the  motion  will  still  take  place  in  the  same 
straight  line;  but  the  velocity  will  vary  as  long  as  the  force 
acts,  and,  from  what  we  have  seen,  the  equation 

F=m*±  (i) 

dt2 

will  hold. 

If  an  unresisted  force  act  in  a  direction  inclined  to  the 
body's  motion,  it  will  cause  the  body  to  change  its  speed,  and 
also  its  course,  and  hence  to  move  in  a  curved  line.  Indeed, 
if  a  force  acting  on  a  body  which  is  in  motion  be  resolved  into 
two  components,  one  of  which  is  tangent  to  its  path  and  the 
other  normal,  the  tangential  component  will  cause  the  body  to 
change  its  speed,  and  the  normal  component  will  cause  it  to 
change  the  direction  of  its  motion. 

The  measure  of  the  tangential  component  is,  as  we  have 
seen, 


and  we  will  proceed  to  find  an  expression  for  the  normal  com- 
ponent  otherwise  known  as  the  Deviating  Force.      For  this 


CENTRIFUGAL   FORCE.  8  1 

purpose  we  may  substitute,  for  a  small  portion  of  the  curve,  a 
portion  of  the  circle  of  curvature  ;  hence  we  will  proceed  to 
find  an  expression  for  the  centrifugal  force  of  a  body  which 
moves  uniformly  with  a  velocity  v  in  a  circle  whose  radius  is  r. 

CENTRIFUGAL    FORCE. 

Let  AC  (Fig.  47)  be  the  space  described  in  the  time  A/. 

Then  we  have  A  B 

AC  = 


The  motion  AC  may  be  approximately  consid- 
ered as  the  result  of  a  uniform  motion 

AB  =  z/A/  nearly, 
.and  a  uniformly  accelerated  motion  PIG.  47. 

BC  =  itf(A/)2  =  s, 
where  a  =  acceleration  due  to  centrifugal  force.     But 

(AB)2  =  BC  .  BD, 
or 

(vkty  =  %a(MY(2r  +  s)t 
where 

AO  =  OC  =  r 

/.    v2     =  %a(2r  +  s)  approximately 

2V2 

.*.     a       =  --  approximately. 

2r  +  s 

For  its  true  value,  pass  to  the  limit  where  s  =  o. 

Hence  we  have,  for  the  acceleration  due  to  the  centrifugal 
force,  the  expression 

£ 
r' 

Hence  the  centrifugal  force  is  -equal  to 


gr 


82  APPLIED  MECHANICS. 


DEVIATING    FORCE. 

If  a  body  is  moving  in  a  curved  path,  whether  circular  or 
not,  and  the  unbalanced  force  acting  on  it  be  resolved  into  tan- 
gential and  normal  components,  the  tangential  component  will 
be,  as  has  already  been  seen, 


and  the  normal  component  will  be 

mv2  _  m/dsV 
r      '-  \dt)' 

where  r  is  the  radius  of  curvature  of  the  path  at  the  point  in 
question. 

RESULTANT    FORCE. 

Hence  it  follows  that  the  entire  unbalanced  force  acting  on 
the  body  will  be 


or 

F  =  m 


§  74.  Components  along  Three  Rectangular  Axes  of  the 
Velocities    of,   and    of    the    Forces    acting    on,   a    Moving 

Rociy.  —  If  we  resolve  the  velocity—  into  three  components 

along  OX,  OY,  and  OZ,  we  shall  have,  for  these  components 
respectively, 

dx       dy  ,      dz 

-         aDd     ' 


this  being  evident  from  the  fact  that  dx,  dy,  and  dz  are  respec- 


COMPONENTS  OF   VELOCITIES  AND   FORCES.  83 

tively  the  projections  of  ds  on  the  axes  OX,  OY,  and  OZ ;  and, 
from  the  differential  calculus,  we  have 


ds_ 
dt 

On  the  other  hand, 


dx       dy          A     dz 

*'  it'  and  7/ 


are  not  only  the  components  of  the  velocity  —  in  the  directions 

OX,  OY,  and  OZ,  but  they  are  also  the  velocities  of  the  body 
in  these  directions  respectively. 

Now,  the  case  of  the  accelerations  is  different  ;  for,  while 


d2x      d2y          ,     d2z 
- 


are  the  accelerations  in  the  directions  OX,  OY,  and  OZ  respec- 
tively, they  are  not  the  components  of  the  acceleration 


dt2 
along  the  three  axes. 

That  they  are  the  former  is  evident  from  the  fact  that  —  , 

dt 

-f-,  and  —  are  the  velocities  in  the  directions  of  the  axes,  and 
at  at 

d2x   d2v   d2z 

—  ,  ~~,  —  are  their  differential  co-efficients,  and  hence  repre- 

sent the  accelerations  along  the  three  axes.     But  if  we  consider 
the  components  of  the  force  acting  on  the  body,  we  shall  have 


84  APPLIED   MECHANICS. 

for  its  components  along  OXt  OY,  and  OZ,  if  a,  ft,  and  y  are 
the  angles  made  by  F  with  the  axes  respectively, 

Fcosa  =  m—,      F  cos  ft  =  m— ^       Fcosy  =  m — -. 
dt2  dt2  ^2 

.-.     F 


and  we  found  (§  73)  for  F,  the  value 


Hence,  equating  these  values  of  F,  and  simplifying,  we  shall 
have  the  equation 


Hence  it  is  plain  that  -— ,  — ^-,  and  — -   can  only  be  the  com- 
dt2    df  dt2 

ponents  of  the  actual  acceleration 

when  the  last  term  — f  —  J  vanishes,  or  when  r  =  oo ,  i.e.,  when 


the  motion  is  rectilinear. 

Moreover,  we  have  the  two  expressions  (i)  and  (2)  for  the 
force  acting  upon  a  moving  body. 

The  truth  of  the  proposition  just  proved  may  also  be  seen 
from  the  following  considerations  :  — 

If  a  parallelopiped  be  constructed  with  the  edges 

dx      dy      dz 


CENTRIFUGAL   FORCE   OF  A    SOLID   BODY.  85 

the  diagonal  will  be  the  actual  velocity 

ds 
df 

and  will,  of  course,  coincide  in  direction  with  its  path. 

On  the  other  hand,  if  a  parallelepiped  be  constructed  with 

the  edges 

d2x       d2y       d2z 
dt2'      dt2'      dt2' 

its  diagonal  must  coincide  in  direction  with  the  force 


and  can  coincide  in  direction  with  the  path,  and  hence  with  the 
actual  acceleration 

d2s 

dt2' 

only  when  the  force  is  tangential  to  the  path,  and  hence  when 
the  motion  is  rectilinear. 

§75.  Centrifugal  Force  of  a  Solid  Body.  —  When  a  solid 
body  revolves  in  a  circle,  the  resultant  centrifugal  force  of  the 
entire  body  acts  in  the  direction  of  the  perpendicular  let  fell 
from  the  centre  of  gravity  of  the  body  on  the  axis  of  rotation, 
and  its  magnitude  is  the  same  as  if  its  entire  weight  were  con- 
centrated at  its  centre  of  gravity. 

PROOF.  —  Let  (Fig,  48)  the  angular  velocity  =  a,  and  the  *eta' 
weight  =  W.     Assume  the  axis  of  rotation  perpendicular  t 
the  plane  of  the  paper  and  passing  through 
O  ;  assume,  as  axis  of  ;r,  the  perpendicular 
dropped  from  the  centre  of  gravity  upon 
the  axis  of  rotation.     The  co-ordinates  of 
the  centre  of  gravity  will  then  be  (r0,  _^0), 
and  y0  will  be  equal  to  zero. 

FIG     8 

If,  now,  P  be  any  particle  of  weight  w, 
where  r  =  perpendicular  distance  from  P  on  axis  of  rotatsoo, 


86  APPLIED   MECHANICS. 

and  x  —  OA,  y  =  AP,  we  shall  have  for  the  centrifugal  force 
of  the  particle  at  P 


w  , 
-a.2r; 

g 


but  if  we  resolve  this  into  two  components,  parallel  respectively 
to  OX  and  OY,  we  shall  have  for  these  components 


and  o.',       =   -wy, 

g     sr       g  \g      /r       g  ' 

and,  for  the  resultant  for  the  entire  body  we  shall  have,  parallel 
to  OX, 

(i) 


g 

and 

Fy  —  — 2wy   =  —  WyQ  =  o.  (2) 

g  g 

Hence  the  centrifugal  force  of  the  entire  body  is 

F-*-W*.;  (3) 

ani  if  we  let  v0  =  o,x0  =  linear  velocity  of  the  centre  of  gravity, 
we  have 

F-  Wv* 

•*•  ~~~          ) 

wnuh  13  the  same  as  though   the  entire  weight  of  the  body 
;cic  concentrated  at  its  centre  of  gravity. 

EXAMPLES. 

H.  A  lo-pound  weight  is  fastened  by  a  rope  5  feet  long  to  the 
centre,  aroun  1  which  it  revolves  at  the  rate  of  200  turns  per  minute ; 
hrd  the  pull  on  the  cord. 

2.  A  locomotive  weighing  50000  Ibs.,  whose  driving-wheels  weigh 
toe  Ibs.,  is  running  at  60  miles  per  hour,  the  diameter  of  the  drivers 


UNIFORMLY   VARYING   RECTILINEAR   MOTION.  8/ 

being  6  feet,  and  the  distance  from  the  centre  of  the  wheel  to  the  centre 
of  gravity  of  the  same  being  2  inches  (the  drivers  not  being  properly 
balanced)  \  find  the  pressure  of  the  locomotive  on  the  track  (a)  when 
the  centre  of  gravity  is  directly  below  the  centre  of  the  wheel,  and  (b) 
when  it  is  directly  above. 

3.  Assume  the  same  conditions,  except-  that  the  distance  between 
•centre  of  the  wheel  and  its  centre  of  gravity  is  5  inches  instead  of  2. 

§76.  Uniformly  Varying  Rectilinear  Motion.  —  We  have 

already  found  for  this  case  (§  68)  the  equations 

—  -  =  a  =  a  constant. 
(it* 


and  we  may  write  for  the  force  acting,  which  is,  of  course,  coin- 
cident in  direction  with  the  motion, 

F  =  m  —  =  ma  =  a  constant. 

dr 

§  77.  Motion  of  a  Body  acted  on  by  the  Force  of  Gravity 
only.  —  A  useful  special  case  of  uniformly  varying  motion  is 
that  of  a  body  moving  under  the  action  of  gravity  only. 

The  downward  acceleration  due  to  gravity  is  represented  by 
g  feet  per  second,  the  value  of  g  varying  at  different  points  on 
the  surface  of  the  earth  according  to  the  following  law  :  — 

g  =  gi(i  —  0.00284  cos  2X)(i  —  —  ^  feet  per  second, 

where 

g,  =  32.1695  feet, 

A   =  latitude  of  the  place, 

•h   =  its  elevation  above  mean  sea-level  in  feet, 

R  —  20900000  feet. 


88  APPLIED   MECHANICS. 

If,  now,  we  represent  by  h  the  height  fallen  through  by  a 
descending  body  in  time  /,  we  shall  have  the  equations, 

v.—  v0  +  gt, 
h  =  vQt  +  \gt*, 

where  v0  is  the  initial  downward  velocity. 

If,  on  the  other  hand,  we  represent  by  v0  the  initial  upward 
velocity,  and  by  h  the  height  to  which  the  body  will  rise  in 
time  /  under  the  action  of  gravity  only,  we  must  write  the  equa- 
tions 


When  v0  =  o,  the  first  set  of  equations  gives 

v  =  gty 
h  =  &/», 

which  express  the  law  of  motion  of  a  body  starting  from  rest 
and  subject  to  the  action  of  gravity  only. 

Eliminate  /  between  these  equations,  and  we  shall  have 


or 


h  is  called  the  height  due  to  the  velocity  v,  and  represents  the 
height  through  which  a  falling  body  must  drop  to  acquire  the 
velocity  v  ;  and 

v  =  \2gh 


UNRESISTED   PROJECTILE.  89 


is  the  velocity  which  a  falling  body  will  acquire  in  falling 
through  the  height  h.  Thus,  if  a  body  fall  through  a  height  of 
50  feet,  it  will,  by  that  fall,  acquire  a  velocity  of  about 


V2(32i)  (5°)  =  V32i6.66  =  56.7  feet  per  second. 

Again  :  if  a  body  has  a  velocity  of  40  feet  per  second,  we  shall 
have 

v2       1600  0  r    , 

h  =  —  —  -    -  =  24.8  feet ; 
*g       64.3 

and  we  say  that  the  body  has  a  velocity  due  to  the  height  24.8 
feet,  i.  e.,  a  velocity  which  it  would  acquire  by  falling  through  a 
height  of  24.8  feet. 


EXAMPLES. 

1.  A  stone  is  dropped  down  a  precipice,  and  is  heard  to  strike  the 
bottom  in  4  seconds  after  it  started  :  how  high  is  the  precipice  ? 

2.  How  long  will  a  stone,  dropped  down  a  precipice  500  feet  high, 
take  to  reach  the  bottom  ? 

3.  What  will  be  its  velocity  just  before  striking  the  ground? 

4.  A  body  is  thrown  vertically  upwards  with  a  velocity  of  100  feet 
per  second  ;  to  what  height  will  it  rise  ? 

5.  A  body  is  thrown  vertically  upwards,  and  rises  to  a  height  of  50 
feet.     With  what  velocity  was  it  thrown,  and  how  long  was  it  in  its 
ascent  ? 

6.  What  will  be  its  velocity  in  its  ascent  at  a  point  15  feet  above 
the   point  from  which   it   started,  and  what   at   the  same  point  in  its 
descent  ? 

§  78.  Unresisted  Projectile.  —  In  the  case  of  an  unresisted 
projectile,  we  have  a  body  on  which  is   impressed  a  uniform 


APPLIED   MECHANICS. 


motion  in  a  certain  direction  (the  direction  of  its  initial  motion), 
and  which  is  acted  on  by  the  force  of  gravity  only. 

Let  OPC  be 
the  path  (Fig.  49), 
OA  the  initial  di- 
rection, and  v0  the 
initial  velocity,  and 
the  angle -4  CUT  = 

K  e. 

Then  we  shall 

FlG  49  have,  for  the  hori- 

zontal and  vertical 
components  of  the  unbalanced  force  acting,  when  the  projectile 
is  at  P  (co-ordinates  x  and  j), 

m  —  =  o  along  OX,  and  m  —  =  —  mg  —  —W  along  O  Y. 
dP  dt* 

Hence 

^  =  °'  ^  Tip  =  ~g'  ^ 

Integrating,  and  observing,  that,  when  t  —  o,  the  horizontal 
and  the  vertical  velocities  were  respectively  z;0cos  0  and  z>0sin  0, 
we  have 

dx  n  ,  , 

—  =  VQ  cos  0,  (3) 

^  •     n  t    \ 

i- ***'-*•  W 

These  equations  could  be  derived  directly  by  observing  that 
the  horizontal  component  of  the  initial  velocity  is  VQ  cos  0,  and 
that  this  remains  constant,  as  there  is  no  unbalanced  force  act- 
ing in  this  direction,  also  that  v0  sin  9  is  the  initial  vertical 
velocity  ;  and,  since  the  body  is  acted  on  by  gravity  only,  this 
velocity  will  in  time  /  be  decreased  by  gt. 


UNRESISTED   PROJECTILE.  91 

Integrating  equations  (3)  and  (4),  and   observing  that  for 
/  —  o,  x  and  y  are  both  zero,  we  obtain 

X  =   VQ  COS  O.t,  (5) 

y  =  VQ  sin  O.t  -  \gt\  (6) 

Eliminate  /,  and  we  have 

^  =  *tan0  --  $*  -  (7) 

2V0*  COS2  0 

as  the  equation  of  the  path,  which  is  consequently  a  parabola. 

Equations  (i),  (2),  (3),  (4),  (5),  (6),  and  (7)  enable  us  to  solve 
any  problem  with  reference  to  an  unresisted  projectile. 

Equation  (7)  may  be  written 


/         v02  sin2  0\  g         / 

V         ~~'  ~~      ~  2Vo*ca*0  \ 


P0»sin0cos0 


which  gives  for  the  co-ordinates  of  the  vertex 

_  v02  sin2  0  _  z/o2  sin  0  cos  0 

y\  ~~~  )  x\  —  -  —  •-  •• 

2g  g 


EXAMPLES. 

i.  An  unresisted  projectile  starts  with  a  velocity  of  100  feet  pei* 
second  at  an  upward  angle  of  30°  to  the  horizon  ;  what  will  be  its  velocity 
when  it  has  reached  a  point  situated  at  a  horizontal  distance  of  icou  teet 
from  its  starting-point,  and  how  long  will  be  required  for  it  to 
that  point? 

Solution. 

v0  =  100,        0  =  30°,        v0  cos  0  =  86.6,        v0  sis  0  as  50, 

g  =  32-i6. 
Equation  (5)  gives  us 

1000  =  86.6  / 

.'.     /  =  =  11.55  seconds. 

86,6 


92 


APPLIED   MECHANICS. 


e>0sin<9  -  gt  =  50  -  371.5  =  -32I-5> 


v  =  V^(86.6)2  +  (32I-5)2  =  V75°°  +  103362  =  333. 

Hence  the  point  in  question  will  be  reached  in   nj  seconds  after  start- 
ing, and  the  velocity  will  then  be  333  feet  per  second. 

2.  An  unresisted  projectile  is  thrown  upwards  from  the  surface  of 
the  earth  at  angle  of  39°  to  the  horizontal :  find  the  time  when  it  will 
reach  the  earth,  and  the  velocity  it  will  have  acquired  when  it  reaches 
the  earth,  the  velocity  of  throwing  being  30  feet  per  second. 

3.  A  lo-pound  weight  is  dropped  from  the  window  of  a  car  when 
travelling  over  a  bridge  at  a  speed  of  25  miles  an  hour.     How  long  will 
it  take  to  reach  the  ground  100  feet  below  the  window,  and  what  will  be 
the  kinetic  energy  when  it  reaches  the  ground  ? 

4.  With  what  horizontal  velocity,  and  in  what  direction,  must  it  be 
thrown,  in  order  that  it  may  strike  the  ground  50  feet  forward  of  the 
point  of  starting? 

5.  Suppose  the  same  lo-pound  weight  to  be  thrown  vertically  up- 
wards from  the  car  window  with  a  velocity  of  100  feet  a  minute,  how 
long  will  it  take  to  reach  the  ground,  and  at  what  point  will  it  strike  the 
ground  ? 

§  79.  Motion  of  a  Body  on  an  Inclined  Plane  without 

Friction.  —  If  a  body  move  on 
an  inclined  plane  along  the  line 
of  steepest  descent,  subject  to 
the  action  of  gravity  only,  and 
if  we  resolve  the  force  acting 
on  it  (i.e.,  its  weight)  into  two 
components,  along  and  perpen- 
dicular to  the  plane  respec- 
tively, the  latter  component 
will  be  entirely  balanced  by 
the  resistance  of  the  plane, 

and  the  former  will   be  the  only  unbalanced   force  acting  on 

the  body. 


MOTION  OF  A   BODY  ON  AN  INCLINED   PLANE.  93 

Suppose  a  body  whose  weight  is  represented  (Fig.  50)  by 
HF  =  W  to  move  along  the  inclined  path  AB  under  the  action 
of  gravity  only.  Let  9  be  the  inclination  of  AB  to  the  horizon. 
Resolve  W  into  two  components, 


and      HE  =  ^cos  9, 


respectively  parallel  and  perpendicular  to  the  plane.  The 
former  is  the  only  unbalanced  force  acting  on  the  body,  and 
will  cause  it  to  move  down  the  plane  with  a  uniformly  accel- 
erated motion  ;  the  acceleration  being 


(i) 


If  the  body  is  either  at  rest  or  moving  downwards  at  the 
beginning,  it  will  move  downwards ;  whereas,  if  it  is  first  mov- 
ing upwards,  it  will  gradually  lose  velocity,  and  move  upwards 
more  slowly,  until  ultimately  its  upward  velocity  will  be  de- 
stroyed, and  it  will  begin  moving  downwards. 

The  equations  for  uniformly  varying  motion  are  entirely 
applicable  to  these  cases.  Thus,  suppose  that  the  body  has  an 
initial  downward  velocity  vot  this  velocity  will,  at  the  end  of  the 
time  /,  become 

z>  =  ^  =  z>0  +  Crsintf)/  (2) 

at 

.-.     s  =  v0t  -f  k  sin  B  .  t*,  (3) 

and,  for  the  unbalanced  force  acting,  we  have 

F=m£l  =  !£(gsmO)  =  WsmO.  (4) 

at2          g 


94  APPLIED  MECHANICS. 

If,  on  the  other  hand,  the  body's  initial  velocity  is  upward, 
and  we  denote  this  upward  velocity  by  vof  we  shall  have  the 
equations 

v  =|=  v0  -  (g*m$)t  (5) 

s   =  v«t  -  fesinfl  ./2  (6) 

F=  -WsinO.  (7) 

Again,  if  the  initial  velocity  is  zero,  equations  (2)  and  (3) 
become 

(8) 


From  these  we  obtain,  for  this  case, 


2S 


do) 


and,  substituting  this  value  of  /  in  (8),  we  have 


v  =  \2g(s  sin  6),  (n) 

or,  if  we  let  s  sin  6  =  h  —  the  vertical  distance  through  which 
the  body  has  fallen,  we  have 


v  —     2gh.  (12) 

Hence,  When  a  body,  starting  from  rest,  falls,  under  the 
action  of  gravity  only,  through  a  height  h,  the  velocity  acquired 
is  \/2gh,  whether  the  path  be  vertical  or  inclined. 


EXAMPLES. 


i.  A  body  moves  from  the  top  to  the  bottom  of  a  plane  inclined 
to  the  horizon  at  30°,  under  the  action  of  gravity  only  :  find  the  time 
required  for  the  descent,  and  the  velocity  at  the  foot  of  the  plane. 


MOTION  ALONG  A    CURVED   LINE. 


95 


FIG.  51. 


2.  In  the  right-angled  triangle  shown  in  the  figure  (Fig.  51),  given 
AB  =  10  feet,  angle  BAC  =  30°:    find  the  time  a  A 
body  would  require,  if  acted  on  by  gravity  only,  to  fall 

from  rest  through  each  of  the  sides  respectively,  AB 
being  vertical. 

3.  Given  inclination  of  plane  to  the  horizon  =  0, 
length  of  plane  =  /.•  compare  the  time  of  falling  down 
the  plane  with  the  time  of  falling  down  the  vertical. 

4.  A  loo-pound  weight  rests,  without  friction,  on  the 
plane  of  example  3.     What  horizontal  force  is  required 
to  keep  it  from  sliding  down  the  plane. 

5.  Suppose  5  pounds  horizontal  force  to  be  applied 

(a)  so  as  to  oppose  the  descent,  (£)  so  as  to  aid  the  descent  :  find  in 
each  case  how  long  it  will  take  the  weight  to  descend  from  the  top  to 
the  bottom  plane. 

§  80.  Motion  along  a  Curved  Line  under  the  Action  of 
Gravity  only.  —  We  shall  consider  two  questions  in  this 
regard  :  (a)  the  velocity  at  any  point  of  the  curve  (b)  the  time 
of  descent  through  any  part  of  the  curve. 

(a)  Velocity  at  any  point.     Let  us  suppose  the  body  to  have 

started  from  rest  at  A,  and  to  have 
reached  the  point  P  in  time  /, 
where  AB  =  x  (Fig.  52).  Then, 
since  the  curved  line  AP  may  be 
considered  as  the  limit  of  a  broken 
line  running  from  A  to  P,  and  as 
it  has  already  been  seen  that  the 
velocity  acquired  by  falling  through 
c  a  certain  height  depends  only  upon 
the  height,  and  not  upon  the  incli- 
nation of  the  path,  we  shall  have  for  a  curved  line  also 


FIG.  52. 


where  v  is  the  velocity  at  P. 


APPLIED   MECHANICS. 


(b)  Time  down  a  curve.  Referring  to  the  same  figure,  let  / 
denote  the  time  required  to  go  from  A  to  P,  and  &t  the  time  to 
go  from  P  to  f,  where  PP'  =  AJ,  and  BB!  =.  kx  ;  then,  as  we 
have  seen  that  the  velocity  at  P  is  \2gx,  we  shall  have  approx- 
imately for  the  space  passed  over  in  time  A/,  the  equation 


or,  passing  to  the  limit, 


This  equation  gives 


tis 


ds 


or 


/  =  c      =  r 

J^2gX  J 


(2) 


v/here,  of  course,   the   proper  limits   of   integration   must   be 
used. 

If  /  denote  the  time  from  A  to  P,  we  have 


=    (""-*= 
J  *    ,,VV 


FK,  53 


EXAMPLE. 


A  body  acted  on  by  gravity  only  is  constrained  to 
move  in  the  arc  of  a  circle  from  A  to  C  (Fig.  53),  radius 
10  feet.  Find  the  time  of  describing  the  arc  (quadrant) 
and  the  velocity  acquired  by  the  body  when  it  reaches 


SIMPLE    CIRCULAR   PENDULUM. 


97 


§8i.   Simple  Circular  Pendulum.  —  To  find  the  time  occu- 
pied in  a  vibration  of  a  simple  circu-  ^c 
lar  pendulum,  we  take  D  (Fig.  54)  as 
origin,  and  DC  as  axis  of  x,  and  the 
axis  of  jj/at  right  angles  to  DC.     Let 
AC  —  /and  BD  =  //,  we  shall  have 
for  the  time  of  a  single  oscillation 
trom  A  to  E 


/-,    f 

J     *  = 


Now,  from  the  equation  of  the  circle  AFDE, 

y2  =    2/X  —  X2, 


we  have 


dy_  =  I  -  x 
dx          y 

ds        I 


y       s/2  ix  -  & 


Idx 


-  x2)\_2g{h  -  *)] 


dx 


-  x2  V2/- 


or 


This  can  only  be  integrated  approximately. 
Expanding  f  i  —  —  J       we  obtain 


(-3T- 


£7  +  ~Ta 
4/       32  /2 


98  APPLIED  MECHANICS. 

The  greatest  value  of  x  is  //;  and  if  h  is  so  small  that  we  may 
omit  — ,  we  shall  have  as  our  approximate  result 

t  =  J-f  /  dx     =  \mvQ™~*^T\k  =  nA    o> 

*  g  J      yhx  —  x2        V  g(  h  )  o  V  ^ 

o 

If,  however,  the  value  of  h  as  compared  with  /  is  too  large 

to  render  it  sufficiently  accurate  to  omit  — ,  but  so  small  that 

4/ 

we  can  safely  omit  the  higher  powers  of  ^,  we  shall  have 

xdx 
h     '    4/t 


h         4/[_2 
or 

' = V^1 + ^          (2) 

a  nearer  approximatioa 
The  formula 


is  the  most  used,  and  is  more  nearly  correct,  the  smaller  the 
value  of  h. 

EXAMPLES. 

i .  Find  the  length  of  the  simple  circular  pendulum  which  is  to  beat 
seconds  at  a  place  where  g  =  32^. 

Solution. 


SIMPLE    CYCLOIDAL   PENDULUM. 


99 


2.  What  is  the  time  of  vibration  of  a  simple  circular  pendulum  5 
feet  long? 

§82.  Simple  Cycloidal  Pendulum.  —  The  equation  of  the 
cycloid  is 

— x  x 

y  =  #  versin      — \-  (2ax  —  Jf3)^ 
a 

.    dy_  —  \/2a  ~  x 

dx       V       x 

ds_    =    /2^\5 

dx       \  x  I 

Hence  we  shall  have,  for  the  time  of  a  single  oscillation, 

dx 


or 


This  expression  is  independent  of  //,  so  that  the  time  of  vibra- 
tion is  the  same  whether  the  arc  be  large  or  small. 

A  body  can  be  made  to  vibrate  in  a  cycloidal  arc  by  suspend- 
ing it  by  a  flexible  string  between  two  cycloidal  cheeks.  This 
is  shown  from  the  fact  that 
the  evolute  of  the  cycloid  is 
another  cycloid  (Fig.  55). 

To  prove  this,  we  have, 
from  the  equation  of  the 
cycloid, 

y  =  a versin      -  -j-  (2ax  — 


dy  _  t  / 
dx  ~  V 


2a  —  x    ds 


—a 


&.^_     

<&       *^ia  -  x 


I00  APPLIED   MECHANICS. 

Hence  the  radius  of  curvature  is 


and  since  we  have  for  the  evolute  the  relation 

ds'  =  dp, 
where  dsf  is  the  elementary  arc  of  the  evolute, 

f*x  =  za 

.-.    /=    I    *; 

i/.*r^* 

and,  observing  that  when   x  —  2a     p  =  o,    we  have 


If  xl  is  the  abscissa  of  the  point  of  the  evolute, 

-  -  x  +    dy  =     a  -  x 
ds 


and,  transforming  co-ordinates  to  B  by  putting  x2.  +  2a  for 
we  obtain 


which  is  the  equation  of  another  cycloid  just  like  the  first. 

The  motion  along  a  vertical  cycloid  may  also  be  obtained  by 
letting  a  body  move  along  a  groove  in  the  form  of  a  cycloid 
acted  on  by  gravity  alone  ;  and  in  this  case  the  time  of  descent 
of  the  body  to  the  lowest  point  is  precisely  the  same  at  what- 
ever point  of  the  curve  the  body  is  placed. 

§  83.  Effect  of  Grade  on  the  Tractive  Force  of  a  Rail- 
way Train.  —  -Asa  useful  particular  case  of  motion  on  an 
inclined  plane,  we  have  the  case  of  a  railroad  train  moving  up 
or  down  a  grade.  It  is  necessary  that  a  certain  tractive  force 


EFFECT  OF  GRADE   ON   TRACTIVE   FORCE.  IOI 

be  exerted  in  order  to  overcome  the  resistances,  and  keep 
the  train  moving  at  a  uniform  rate  along  a  level  track.  If, 
on  the  other  hand,  the  track  is  not  on  a  level,  and  if  we 
resolve  the  weight  of  the  train  into  components  at  right  angles 
to  and  along  the  plane  of  the  track,  we  shall  have  in  the  latter 
component  a  force  which  must  be  added  to  the  tractive  force 
above  referred  to  when  we  wish  to  know  the  tractive  force  re- 
quired to  carry  it  up  grade,  and  must  be  subtracted  when  we 
wish  to  know  the  tractive  force  required  to  carry  it  down  grade. 
The  result  of  this  subtraction  may  give,  if  the  grade  is  suffi- 
ciently steep  and  the  speed  sufficiently  slow,  a  negative  quan- 
tity ;  and  in  that  case  we  must  apply  the  brakes,  instead  of 
using  steam,  unless  we  wish  the  speed  of  the  train  to  increase. 

EXAMPLES. 

i.  A  railroad  train  weighing  60000  Ibs.,  and  running  at  50  miles  per 
hour,  requires  a  tractive  force  of  618  Ibs.  on  a  level ;  what  is  the  tractive 
force  necessary  when  it  is  to  ascend  a  grade  of  50  feet  per  mile?  What 
when  it  is  to  descend?  Also  what  is  the  amount  of  work  per  minute 
in  each  case  ? 

Solution. 

The  resolution  of  the  weight  will  give  (Fig.  50,  §  7?^  tor  the  com- 
ponent along  the  plane, 

(60000)^  =  568.2  nearly. 
Hence 

Tractive  force  for  a  level  =  618.0, 
Tractive  force  for  ascent  =  1186.2, 
Tractive  force  for  descent  —  49.8. 

To  ascertain  the  work  done  per  minute  in  each  case,  we  have  — 
(a)   For  a  level  track,  6l8  x  56°ox  528°  =  2719200  foot-lbs. 

(l>)  Up  grade,       2719200  +  6ooo°  ^  x  5°  =  5219200  foot-lbs. 
(c)   Down  grade,  2719200  -  6ooo°  XJ°  x  5°  =     219200  foot-lbs. 


102 


APPLIED   MECHANICS, 


2.  Suppose  the  tractive  force  required  for  each  2000  Ibs.  of  weight 
of  train  to  be,  on  a  level  track,  for  velocities  of — 

5.0  miles  per  hour,     10.0     20.0     30.0     40.0     50.0     60 

6.1  Ibs.,  6.6       8.3     ii. 2     15.3     20.6     27; 
find  the  tractive  force  required  to  carry  the  train  of  example  i  — 

(a)  Up  an  incline  of  50  feet  per  mile  at  30  miles  per  hour. 
(^)   Down  an  incline  of  50  feet  per  mile  at  30  miles  per  hour. 
(<:)   Down  an  incline  of  10  feet  per  mile  at  20  miles  per  hour. 
(//)   What  must  be  the  incline  down  which  the  train  must  run  to 
require  no  tractive  force  at  40  miles  per  hour? 

3.  If  in  the  first  example  the  tractive  force  remains  618  Ibs.  while 
the  train  is  going  down  grade,  what  will  be  its  velocity  at  the  end  of  one 
minute,  the  grade  being  10  feet  per  mile? 

§84.    Harmonic    Motion If  we   imagine  a  body   to  be 

moving  in  a  circle  at  a  uniform  rate  (Fig.  56),  and  a  second 

body  to  oscillate  back  and  forth  in 
the  diameter  AB,  both  starting 
from  B,  and 
if  when  the 
first  body  is 
•*  at  C  the  other 
is  directly  un- 
der it  at  G, 
etc.,  then  is 
the  second 
body  said  to 


FIG.  56. 


move  in  harmonic  motion. 

A  practical  case  of  this  kind  of  mo- 
tion is  the  motion  of  a  slotted  cross-head 
of   an  engine,   as  shown   in    the   figure 
ig-  57)  i    the  crank  moving  at  a 


form  rate.  In  the  case  of  the  ordinary 
crank,  and  connecting-rod  connecting 
the  drive-wheel  shaft  of  a  stationary  engine  with  the  piston-rod, 


FIG.  57. 


HARMONIC  MOTION.  1  03 

we  have  in  the  motion  of  the  piston  only  an  approximation  to 
harmonic  motion.  We  will  proceed  to  determine  the  law  of  the 
force  acting  upon,  and  the  velocity  of,  a  body  which  is  con- 
strained to  move  in  harmonic  motion.  Let  the  body  itself  and 
the  corresponding  revolving  body  be  supposed  to  start  from 
B  (Fig.  56),  the  latter  revolving  in  left-handed  rotation  with  an 
angular  velocity  a,  and  let  the  time  taken  by  the  former  in 
reaching  G  be  t:  then  will  the  angle  BOC  —  at;  and  we  shall 
have,  if  s  denote  the  space  passed  over  by  the  body  that  moves 
with  harmonic  motion, 

s  =  BG  —  OB  -  OCcosat, 
or,  if 

r=O£=  OCt 

s  =  r  —  rcosa/,  (l) 

the  velocity  at  the  end  of  the  time  t  will  be 

V  =  —  =  arsina/,  (2) 

and  the  acceleration  at  the  end  of  time  /  will  be 

(3) 


Hence  the  force  acting  upon  the  body  at  that  instant,  in  the 
direction  of  its  motion,  is 

F  =  m—  =  ma2  r  cos  at  =  ma2(OG).  (4) 

dt* 


The  force,  therefore,  varies  directly  as  the  distance  of  the  body 
from  the  centre  of  its  path.     It  is  zero  when  the  body  is  at  the 


IO4  APPLIED   MECHANICS. 

centre  of  its  path,  and  greatest  when  it  is  at  the  ends  of  its 
travel,  as  its  value  is  then 

W 
ma2r  =  — o?r; 

S 

this  being  the  same  in  amount  as  the  centrifugal  force  of  the 
revolving  body,  provided  this  latter  have  the  same  weight  as  the 
oscillating  body.  On  the  other  hand,  the  velocity  is  greatest 

when  at  =  -  (i.e.,  at  mid-stroke) ;  and  its  value  is  then 

v  =  ar, 
this  being  also  the  velocity  of  the  crank-pin  at  mid-stroke. 

EXAMPLE. 

Given  that  the  reciprocating  parts  of  an  engine  weigh  10000  Ibs., 
the  length  of  crank  being  i  foot,  the  crank  making  60  revolutions  per 
minute  ;  find  the  force  required  to  make  the  cross-head  follow  the  crank, 
(i)  when  the  crank  stands  at  30°  to  the  line  of  dead  points,  (2)  when 
at  60°,  (3)  when  at  the  dead  point. 

§85.  Work  under  Oblique  Force.  —  If  the  force  act  in 
any  other  direction  than  that  of  the  motion,  we  must  resolve  it 
into  two  components,  the  component  in  the  direction  of  the 
motion  being  the  only  one  that  does  work.  Thus  if  the  force 
F  is  variable,  and  6  equals  the  angle  it  makes  with  the  direction 
of  the  motion,  we  shall  have  as  our  expression  for  the  work 
done 

fFcosOds. 

Thus  if  a  constant  force  of  100  Ibs.  act  upon  a  body  in  a  direc- 
tion making  an  angle  of  30°  with  the  line  of  motion,  then  wil! 
the  work  done  by  the  force  during  the  time  in  which  it  moves 
through  a  distance  of  10  feet  be 

(100)  (0.86603)  (10)  =  866  foot-lbs. 


ROTATION  OF  RIGID   BODIES.  1  05 

§  86.  Rotation  of  Rigid  Bodies  --  Suppose  a  rigid  body 
(Fig.  58)  to  revolve  about  an  axis  perpendicular  to  the  plane  of 
the  paper,  and  passing  through  O  ; 
imagine  a  particle  whose  weight  is 
w  to  be  situated  at  a  perpendicular 
distance  OA  =  r  from  the  axis  of 
rotation,  and  let  the  angular  accel- 
eration be  a  :  let  it  now  be  required 
to  find  the  moment  of  the  force  or 
forces  required  to  impart  this  ac- 
celeration ;  for  we  know  that,  if 

the  axis  of  rotation  pass  through  the  centre  of  gravity  of  the 
body,  the  motion  can  be  imparted  only  by  a  statical  couple  ; 
whereas  if  it  do  not  pass  through  the  centre  of  gravity,  the 
motion  can  be  imparted  by  a  single  force. 

We  shall  have,  for  the  particle  situated  at  A, 

Weight  =  w. 

Angular  acceleration  =  a. 

Linear    acceleration   =±  o.r. 

Force  required  to  impart  this  acceleration  to  this  particle 

w 

—  —  a-r. 
g 

7ff 

Moment  of  this  force  about  the  axis  =  —  ar2. 

g 

Hence  the  moment  of  the  force  or  forces  required  to  impart 
to  the  entire  body  in  a  unit  of  time  a  rotation  about  the  axis 
through  O,  with  an  angular  velocity  a,  is 


8  8  S 


where  /  is  used  as  a  symbol  to  denote  the  limit  of  ^wr2,  and  is 
called  the  Moment  of  Inertia  of  the  body  about  the  axis  through  O. 


106  APPLIED   MECHANICS. 

§87.  Angular  Momentum.  —  This  quantity,—,  which  ex- 

g 

presses  the  moment  of  the  force  or  forces  required  to  impart  to 
the  body  the  angular  acceleration  a  about  the  axis  in  question 
is  also  called  the  Angular  Momentum  of  the  body  when  rotat- 
ing with  the  angular  velocity  «  about  the  given  axis. 

§  88.  Actual  Energy  of  a  Rotating  Body.  —  If  it  be  re- 
quired to  find  the  actual  energy  of  the  body  when  rotating 
with  the  angular  velocity  w,  we  have,  for  the  actual  energy  of 
the  particle  at  A, 


g         2  2g 

and  for  that  of  the  entire  body 

<u*  w2/ 

—  Iwr2  =  -  . 

*g  zg 

This  is  the  amount  of  mechanical  work  which  would  have  to  be 
done  to  bring  the  body  from  a  state  of  rest  to  the  velocity  w,  or 
the  total  amount  of  work  which  the  body  could  do  in  virtue 
of  its  velocity  against  any  resistance  tending  to  stop  its 
rotation. 

§  89.  Moment  of  Inertia.  —  The  term  "moment  of  inertia" 
originated  in  a  wrong  conception  of  the  properties  of  matter. 
The  term  has,  however,  been  retained  as  a  very  convenient  one, 
although  the  conceptions  under  which  it  originated  have  long 
ago  vanished.  The  meaning  of  the  term  as  at  present  used,  in 
relation  to  a  solid  body,  is  as  follows  :  — 

The  moment  of  inertia  of  a  body  about  a  given  axis  is  the 
limit  of  the  sum  of  the  products  of  the  weight  of  each  of  the  ele- 
mentary particles  that  make  tip  the  body,  by  the  squares  of  their 
distances  from  the  given  axis. 

Thus,  if  wlt  w2,  wy  etc.,  are  the  weights  of  the  particles 
which  are  situated  at  distances  rlf  r»  ry  etc.,  respectively  from 


MOMENT  OF  INERTIA    OF  A    PLANE  SURFACE.  IO/ 

the  axis,  the  moment  of  inertia  of  the  body  about  the  given 

axis  is 

/  =  limit  of 


§  90.  Radius  of  Gyration.  —  The  radius  of  gyration  of  a 
body  with  respect  to  an  axis  is  the  perpendicular  distance  from 
the  axis  to  that  point  at  which,  if  the  whole  mass  of  the  body 
were  concentrated,  the  angular  momentum,  and  hence  the  mo- 
ment of  inertia,  of  the  body,  would  remain  the  same  as  they  are 
in  the  body  itself. 

If  p  is  the  radius  of  gyration,  the  moment  of  inertia  would 
be,  when  the  mass  is  concentrated, 


hence  we  must  have 
whence 


where  W  =  entire  weight  of  the  body. 

§91.  Moment  of  Inertia  of  a  Plane  Surface The  term    , 

"moment  of  inertia,"  when  applied  to  a  plane  figure,  must,  of 
course,  be  defined  a  little  differently,  as  a  plane  surface  has  no 
weight ;  but,  inasmuch  as  the  quantity  to  which  that  name  is 
given  is  necessary  for  the  solution  of  a  great  many  questions. 

The  moment  of  inertia  of  a  plane  surface  about  an  axis,  either 
in  or  not  in  the  plane,  is  the  limit  of  the  sum  of  the  products  of 
the  elementary  areas  into  which  the  surface  may  be  conceived  to 
be  divided,  by  the  squares  of  their  distances  from  the  axis  in 
question. 

In  a  similar  way,  for  the  radius  of  gyration  p  of  a  plane 
figure  whose  area  is  A,  we  have 


108  APPLIED   MECHANICS. 

From  this  definition  it  will  be  evident,  that,  if  the  surface  be 
referred  to  a  pair  of  axes  in  its  own  plane,  the  moment  of  iner- 
tia of  the  surface  about  O  Y  will  be 


(i) 

and  the  moment  of  inertia  of  the  surface  about  OX  will  be 

J^fffdxdy.  (2) 

The  moment  of  inertia  of  the  surface  about  an  axis  passing 
through  the  origin,  and  perpendicular  to  the  plane  XO  Y,  will  be 

SS**dx*y,  (3) 

where  r=.  distance  from  O  to  the  point  (x,y)  ;  hence  r2  =.  x2  -f 
y2,  and  the  moment  of  inertia  becomes 

ff(x2  4-  y2)dxdy  =  ffx2dxdy  +  ffy2dxdy  =  /  +  /.      (4) 

This  is  called  the  "polar  moment  of  inertia."  If  polar  co-ordi- 
nates be  used,  this  last  becomes 

ffp2(PdpdB)  =  ffptdpdO.  (5) 

All  these  quantities  are  quantities  that  will  arise  in  the  discus- 
sion of  stresses,  and  the  letters  /and./  are  very  commonly  used 
to  denote  respectively 

ffxzdxdy        and         ffy*dxdy. 

Another  quantity  that  occurs  also,  and  which  will  be  repre- 
sented by  K,  is 

ffxydxdy;  (6) 

and  this  is  called  the  moment  of  deviation. 


EXAMPLES  OF  MOMENTS   OF  INERl'IA. 


109 


EXAMPLES. 


The  following   examples   will   illustrate   the   mode   of  finding   the 
moment   of   inertia  :  —  .x 

i.  Find  the  moment  of  inertia  of  the  rectangle 
ABCD  about  OY  (Fig.  59). 


Solution. 

h 


FIG.  59. 


2.  Find  the  moment  of  inertia  of  the  entire  circle  (radius  r)  about 
the  diameter  OY  (Fig.  60). 


FIG.  60. 


Solution. 


_ 
4    "  "    64 


=2 


-xtf+r*  f  Vr 

/          4V 


3-  Find  the  moment  of  inertia  of  the  circular  ring  (outside  radius  r, 
inside  radius  r^  about  OY  (Fig.  61). 


Solution. 


"44 


64 


4.  Find  the  moment  of  inertia  of  an  ellipse 
(semi-axes  a  and  b)  about  the  minor  axis  OY. 


FIG.  61. 


no 


APPLIED   MECHANICS. 


Solution. 


Equation  of  ellipse  is  —  -f-  ^ 


7ta*b 


On  the  other  hand,  Ix  — 

4 

§  92.  Moments  of  Inertia  of  Plane  Figures  about  Parallel 
Axes. 

PROPOSITION.  —  The  moment  of  inertia  of  a  plane  figure 
about  an  axis  not  passing  through  its  centre  of  gravity  is  equal 
to  its  moment  of  inertia  about  a  parallel  axis  passing  through  its 
centre  of  gravity  increased  by  the  product  obtained  by  multiply- 
ing the  area  by  the  square  of  the  distance  between  the  two  axes. 

PROOF.  —  Let  A  B  CD 
(Fig.  62)  be  the  surface ;  let 
0Fbe  the  axis  not  passing 
through  the  centre  of  grav- 
ity ;  let  P  be  an  elementary 
area  A^rAr,  whose  co-ordi- 


nates  are  OR  —  x  and  RP 
—  y  ;  and  let  OOT  —  a  =  a 
constant    =    distance    be- 
tween the  axes. 
Let  O,R  —  x,  —  abscissa  of  P  with  reference  to  the  axis 
passing  through  the  centre  of  gravity, 


x  =  a  -f- 
x2  =  x,2 


2axt 


Ay 


POLAR  MOMENT   OF  INERTIA    OF  PLANE  FIGURES.      Ill 


Hence,  summing,  and  passing  to  the  limit,  we  have 

fftfdxdy  =  fjxfdxdy  +  zaffxjxdy  +  a*ffdxdy  ;      (  i  ) 

but  if  we  were  seeking  the  abscissa  of  the  centre  of  gravity 
when  the  surface  is  referred  to  Y^OYlt  and  if  this  abscissa  be 
denoted  by  xm  we  should  have 


_ 
= 


ffdxdy  ' 

and,  since  XQ  =  o,     /.  ffx^dxdy  =  o ;  hence,  substituting   this 
value  in  (l),  we  obtain 

ffx2dxdy  =  ffxfdxdy  -f-  a2 ffdxdy.          .      (2) 

If,  now,  we  call  the  moment  of  inertia  about  O  Y,  7,  that 
about  O,  Ylt  /„  and  let  the  area  =  A  =  ffdxdy,  we  shall  have 

7=7,  +  a- A.  (3) 

Q.  E.  D. 

§  93.  Polar  Moment  of  Inertia  of  Plane  Figures.  —  The 

moment  of  inertia  of  a  plane 
figure  about  an  axis  perpen- 
dicular to  the  plane  is  equal 
to  the  sum  of  its  moments 
of  inertia  about  any  pair  of 
rectangular  axes  in  its  plane 
passing  through  the  foot  of 
the  perpendicular. 

PROOF.  —  Let  BCD  (Fig. 
63)  be  the  surface,  and  P  an    ^Y 
elementary    area,    and    let 
OA  —  x,  AP  =  y,  OP  —  r;   then  the  moment  of   inertia  of 
the  surface  about  OZ  will  be 

f  ffdxdy  =  ff(x2  +y*}dxdy  =  ffx*dxdy  +  f  ffdxdy  =  /  -f  /. 
Q.  E.  D. 


FIG.  63. 


112  APPLIED   MECHANICS. 

Hence  follows,  also,  that  the  sum  of  the  moments  of  inertia 
of  a  plane  surface  relatively  .to  a  pair  of  rectangular  axes  in  its 
own  plane  is  isotropic  ;  i.e.,  the  same  as  for  any  other  pair  of 
rectangular  axes  meeting  at  the  same  point,  and  lying  in  its 
plane. 


EXAMPLES. 


i.  To  find  the  moment  of  inertia  of  the  rectangle  (Fig.  59)  about 
an  axis  through  its  centre  perpendicular  to  the  plane  of  the  rectangle. 


Solution. 
Moment  of  inertia  about  YY  —  — , 

12 

Moment  of  inertia  about  an  axis  through  its 


hence 


centre  and  perpendicular  to  YY  =  — 

12 


Polar  moment  of  inertia  = 1 =  —  (h2  + 

12  12  12 


2.  To  find  the  moment  of  inertia  of  a  circle  about  an  axis  through 
its  centre  and  perpendicular  to  its  pla'ne  (Fig.  60). 


Solution. 

Moment  of  inertia  about  OY  =  — , 

4 


hence 


Moment  of  inertia  about  OX  = • 

4 


-r,    ,  ,   .  Trr4  TIT4  TTf 

Polar  moment  of  inertia  =  -   — f-  —  = 

442 


3.  To  find  the  moment  of  inertia  of  an  ellipse  about  an  axis  passing 
through  its  centre  and  perpendicular  to  its  plane. 


MOMENTS   OF  INERTIA   ABO^JT  DIFFERENT  AXES.      113 


Solution* 
From  example  4,  §91,  we  have 

/   --  7ra^3 
4 


.:     Polar  moment  of  inertia  = (a2 

4 


§  94.  Moments  of  Inertia  of  Plane  Figures  about  Different 
Axes  compared.  —  Given  the  surface  KLM  (Fig.  64),  suppose 
we  have  already  determined  the  quantities 

/  =  ffx2dxdy,  /  =  fffdxdy,   K  =  ffxydxdy, 
it  is  required  to  determine,  in  terms  of  them,  the  quantities 


A 


the  angles  JTOFand  X,OY,  being  both 
right  angles,  and  YO  Y,  =  a. 

We  shall  have,  from  the  ordinary 
equations  for  the  transformation  of  co- 
ordinates, to  be  found  in  any  analytic 
geometry,  the  equations 

xt  =  x  cos  a  -f-  y  sin  a, 

y,  =  ycosa  -  *sina,  FIG  ^ 

x?  =  x2  cos2  a  -f-  y2  sin2  a  -f-  2xy  cos  a  sin  a, 
jj2  =  ^2  sin2  a  -f-  jy2  cos2  a  —  2^'  cos  a  sin  a, 
^jjj  =  ^y(cos2  a  —  sin2  a)  —  (x2  —  y2)  cos  a  since. 


1  14  APPTIED  MECHANICS. 


Hence 


=  ffxfdxdy*  =  limit  of  Sxf&A 
=  cos2  a  limit  of  2x2&A  +  sin2  a  limit  of 
2  cos  a  sin  a  limit 


2  (cos  a  sin  a)  ffxydxdy. 
Js  =  ffyl2dxldyl  =  limit  of  lyf&A 

=  (sin2  a)   limit  of  2^A^  +  (cos2  a)   limit  of 
2  (cos  a  sin  a)  limit  of 


2  (cos  a  sin  a)ffxydxdy. 
Kt  =  ffxly1^xj^yl  =  limit  of  S^ij^iA^ 

=  (cos2  a  —  sin2  a)  limit  of  2<xy&A  —  (cos  a  sin  a)   {limit  of 

2x2&A  -  limit  of  ^y2^A\ 
=  (cos2  a  —  sin2  a)ffxydxdy  —  (cos  a  sin  a)  \ffx2dxdy  — 

fffdxdy}. 

Or,  introducing  the  letters  /,  Jt  and  TsT,  we  have 
7,  =  /cos2  a  +  /  sin2  a  -f  2^  cos  a  sin  a,  (i) 

yr  =  /sin2  a  +  y  COS2  a  —   2  A"  COS  a  sin  a,  (2) 

^  =  (J—  /)  cos  a  sin  a  +  ^(cos2a  —  sin2a).  (3) 

The  equations   (i),   (2),  and  (3)   furnish   the   solution   of    the 
problem. 

§95.  Principal  Moments  of  Inertia  in  a  Plane.  —  In  every 
plane  figure,  a  given  point  being  assumed  as  origin,  there  is  at 
least  one  pair  of  rectangular  axes,  about  one  of  which  the  moment 
of  inertia  is  a  maximum,  and  a  minimum  about  the  other  ;  these 
moments  of  inertia  being  called  principal  moments  of  inertia, 
and  the  axes  about  which  they  are  taken  being  called  principal 
axes  of  inertia 


AXES  OF  SYMMETRY  OF  PLANE   FIGURES.  I  15 

PROOF.  —  In  order  that  /„  equation  (i),  §  94,  may  be  a  maxi- 
mum or  a  minimum,  we  must  have,  as  will  be  seen  by  differen- 
tiating its  value,  and  putting  the  first  differential  co-efficient 
equal  to  zero, 

—  2/cos  a  sin  a  4-  2/cos  a  sin  a  4-  2^(cos2  a  —  sin2  a)  =  o 
/.     ^(cos2  a  —  sin2  a)  —  (/  —  /)  cos  a  sin  a  =  o  (i) 

cos  a  sin  a  K  iK         ,  \ 

/.    -  =  -  .*.     tan  2  a  =  -  -  -.     (2) 

cos2  a  —  sin2  a       /  —  •  J  I  —  J 

Hence,  for  the  value  of  a  given  by  (2),  we  have  7,  a  maximum 
or  a  minimum  ;  and  as  there  are  two  values  of  2a  corresponding 
to  the  same  value  of  tan  2a,  and  as  these  two  values  differ  by 
1  80°,  the  values  of  a  will  differ  by  90°,  one  corresponding  to  a 
maximum  and  the  other  to  a  minimum. 

Moreover,  when  the  value  of  a  is  so  chosen,  we  have 


as  is  proved  by  equation  (i).     Indeed,  we  might  say  that  the 
condition  for  determining  the  principal  axes  of  inertia  is 

K,  =  o. 

§96.  Axes  of  Symmetry  of  Plane  Figures.  —  An  axis 
which  divides  the  figure  symmetrically  is  always  a  principal 
axis. 

PROOF.  —  Let  us  assume  that  the  y  axis  divides  the  surface 
symmetrically  ;  then  we  shall  have,  with  reference  to  this  axis, 


K  = 


And,  since  K  is  zero,  the  axis  of  y  is  one  principal  axis,  and  of 
course  the  axis  of  x  is  the  other.  The  same  method  of  reason- 
ing would  show  K  =  o  if  the  x  axis  were  the  axis  of  symmetry. 


II 6  APPLIED   MECHANICS. 

Hence,  whenever  a  plane  figure  has  an  axis  of  symmetry, 
this  axis  is  one  of  the  principal  axes,  and  the  other  is  at 
right  angles  to  it.  Thus,  for  a  rectangle,  when  the  axis  is  to 
pass  through  its  centre  of  gravity,  the  principal  axes  are  par- 
allel to  the  sides  respectively,  the  moment  of  inertia  being- 
greatest  about  the  shortest  axis,  and  least  about  the  longest. 
Thus  in  an  ellipse  the  minor  axis  is  the  axis  of  maximum, 
and  the  major  that  of  minimum,  moment  of  inertia,  etc.  On 
the  other  hand,  in  a  circle,  or  in  a  square,  since  the  maximum 
and  minimum  are  equal,  it  follows  that  the  moments  of  inertia 
about  all  axes  passing  through  the  centre  are  the  same. 

§  97.  Conditions  for  Equal  Values  of  Moment  of  In- 
ertia.—  When  the  moments  of  inertia  of  a  plane  figure  about 
three  different  axes  passing  through  the  same  point  are  the 
same,  the  moments  of  inertia  about  all  axes  passing  through 
this  point  are  the  same. 

PROOF.  —  Let  /  be  the  moment  of  inertia  about  O  Y,  7l 
about  OYlt  I2  about  OY2,  and  let 

YOY,  =  a,         YOY2  =  ft, 
and  let 

/,  =  /*  =  /. 

Then,  from  equation  (i),  §94,  we  have 

1=1  cos2  a  4-  J  sin2  a  -f  2  K  cos  a  sin  a, 

/=  /cos2/?  +/sin2/3  +  2  A"  cos/?  sin/3. 
Hence 

(/  — y)sin2a   =  2  A' cos  a  sin  a,  (i) 

(7-/)sin2/3  ==  2  K  cos  ft  sin  ft.  (2) 

Hence 

(7-/)tana  =  2AT,  (3) 

(7-/)tan0  =  2  AT.  (4) 

And,  since  tan  a  is  not  equal  to  tan  ft,  we  must  have 
/  —  J  —  o         and         K  =  o. 

Hence,  since  K  —  o  and  /  =  Jt  we  shall  have,  from  eqja- 


MOMENTS   OF  INERTIA    ABOUT  PARALLEL    AXES.        I  I/ 

tion    (i),    §94,   for  the  moment    of   inertia  /'  about    an    axis, 
making  any  angle  0  with  O  Y, 

I'  =  /cos2  0  +  /sin2  0  +  o  =  /.  (5) 

Hence  all  the  moments  of  inertia  are  equal. 

§  98.  Components  of  Moments  of  Inertia  of  Solid 
Bodies.  —  Refer  the  body  to  three  rectangular  axes,  OX,  OY, 
and  OZ ;  and  let  Ix,  Iy,  and  Iz  represent  its  moment  of  inertia 
about  each  axis  respectively.  Then,  if  r  denote  the  distance  of 
any  particle  from  OZ>  we  shall  have 

Iz  =  limit  of  ^wr2  \ 
but 

r*  =  x2  +  y2 

.'.    Iz  =  limit  of  So/C*2  +  y2}  =  limit  of  Saw2  +  limit  of  So/?2,  (i) 
In  the  same  way  we  have 

7*  =  limit  of  ^wy2   +  limit  of  ^wz2,  (2) 

Iy   =  limit  of  Sow2  +  limit  of  ^wz2.  (3) 

§99.  Moments  of  Inertia  of  Solids  around  Parallel 
Axes.  — The  moment  of  inertia  of  a  solid  body  about  an  axis 
not  passing  through  its  centre  of  gravity  is  equal  to  its  moment 
of  inertia  about  a  parallel  axis  passing  through  the  centre  of 
gravity,  increased  by  the  product  of  the  entire  weight  of  the 
body  by  the  square  of  the  distance  between  the  two  axes. 

PROOF.  —  Refer  the  body  to  a  system  of  three  rectangular 
axes,  OX,  OY,  and  OZ,  of  which  OZ  is  the  one  about  which 
the  moment  of  inertia  is  taken.  Let  the  co-ordinates  of  the 
centre  of  gravity  of  the  body  with  reference  to  these  axes  be 
(•^oi  Jo,  #<>)•  Through  the  centre  of  gravity  of  the  body  draw  a 
system  of  rectangular  axes,  parallel  respectively  to  OX,  OY,  and 
OZ.  Then  we  shall  have  for  the  co-ordinates  of  any  point 

X   =    Xo   -\~   Xi, 

y  =  y*  +y» 

Z    =   Z0    +  *,. 


APPLIED   MECHANICS. 


Hence 


72  =  limit  of  2w(x2  +  y2)  =  limit  of  *Zwx2  4  limit  of 
=  limit  of  *2w(x0  4-  x,)2  4-  limit  of  %w(y0  +  yt)2 
=  x02  limit  of  2o>  4-    02  limit  of  2w  4-  2x0  limit  of 

4-  limit  of 


2  limit  of  2o>  4-  y02  limit  of  2w 
-f  2y0  limit  of  Soy^  -f  limit  of 


-f 


limit  of 


=  (*02  -f  7o2) 

-f  limit  of 
=  r02  W  4-  //  4  2^0  limit  of 

But,  since  6^x  is  the  centre  of  gravity, 

/.     ^wxI  =  o         and 
Hence 


4 


limit  of 


limit  of 


o. 


which  proves  the  proposition. 

§  100.  Examples  of  Moments  of  Inertia. 

i  .  To  find  the  moment  of  inertia  of  a  sphere  whose  radius  is  r  and 
weight  per  unit  of  volume  w,  about  the  axis  OZ  drawn  through  its  centre. 

Solution. 

Divide  the  sphere  into  thin  slices  (Fig.  65)  by  planes  drawn  perpen- 

dicular to  OZ.  Let  the  distance 
of  the  slice  shown  in  the  figure, 
above  O  be  z,  and  its  thickness  dz  : 
then  will  its  radius  be  Vr2  —  z2  ; 
and  we  can  readily  see,  from  ex- 
ample  2,  §  93,  that  its  moment  of 
inertia  about  OZ  will  be 


dz. 


FIG.  65. 


Hence  the  moment  of  inertia 
of  the  entire  sphere  about  OZ  will 
be 


w 


-  f  V  - 

2  J_rV 


EXAMPLES  OF  MOMENTS   OF  INERTIA. 


which  easily  reduces  to 


Iz     =  — 
15 


2.  To  find  the  moment  of  inertia  of  an  ellipsoid  (semi-axes  a,  b,  c) 
about  OZ  (Fig.  66). 

SOLUTION. — The  equa- 
tion of  the  ellipsoid  is 


Divide  it  into  thin  slices 
perpendicular  to  OZ,  and 
let  the  slice  shown  in  the 
figure  be  at  a  distance  z 
from  O.  Then  will  this 
slice  be  elliptical,  and  its 
semi-axes  will  be 


FIG.  66. 


-  V<T2    -   Z2 


and 


C*   — 


and  from  example  3,  §  93,  we  readily  obtain,  for  its  moment  of  inertia 
about  OZ, 


= 


Hence,  for  the  moment  of  inertia  of  the  ellipsoid  about   OZ,  we 


have 


I 

U— 


IS 

3.  Find  the  moment  of  inertia  of  a  right  circular  cylinder,  length  a, 
radius  r,  about  its  axis. 

Ans. 


120  APPLIED   MECHANICS. 


4.  Find  the  moment  of  inertia  of  the  same  about  an  axis  perpen- 
dicular to,  and  bisecting  its  axis.  Ans      wwar*  I  a*\ 

4      V        "   3/ 

5.  Find  the  moment  of  inertia  of  an  elliptic  right  cylinder,  length 

2<r,  transverse  semi-axes  a  and  b,  about  its  longitudinal  axis. 


Ans.  ~(a2  +  b2}. 

6.    Find  the  moment  of  inertia  of  the  same  about  its  transverse 
axis  2b. 


Ans. 


+  -)• 

3/ 
7.  Find  the  moment  of  inertia  of  a  rectangular  prism,  sides  2a,  2b, 

2c,  about  central  axis  2C.  Ans.     %wabc(a2  -f  b2}. 

§  101.  Centre  of  Percussion.  —  Suppose  we  have  a  body 
revolving,  with  an  angular  velocity  a,  about  an  axis  perpendicu- 
lar to  the  plane  of  the  paper,  and 
passing  through  O.  Join  O  with 
the  centre  of  gravity,  G,  and  take  OG 
as  axis  of  x ;  the  axis  of  y  passing 
through  (9,  and  lying  in  the  plane  of 
the  paper.  If,  with  a  radius  OA  —  ;-, 
we  describe  an  arc  CA  (Fig.  67),  all 
particles  situated  in  this  arc  have  a 

linear  velocity  o.r.     The  force  which  would  impart  this  velocity 
to  any  one  of  them,  as  that  at  A,  in  a  unit  of  time,  is 


g 

and  this  may  be  resolved  into  two, 

w  ,      w 

—ax     and      —ay, 

S  g 

respectively  perpendicular  and  parallel  to  OG.    The  moment  of 
this  force  about  the  axis  is 


g 

hence  the  total  moment  of  the  forces  which  would  impart  to 


CENTRE   OF  PERCUSSION.  \2l 


the  body  in  a  unit  of  time  the  angular  velocity  a,  is,  as  has  been 
shown  already, 


g         g 
The  resultant  of  the  forces  acting  on  the  body  is 


g 

since,   the   centre   of   gravity   being   on   OB,    it   follows   that 
^wy  =  o  ;  and  hence 

-*2wy  —  o. 
g 

Hence  the  perpendicular  distance  from  O  to  the  line  of  direc- 
tion of  the  resultant  force  is  measured  along  OG,  and  is 


g  7  f  \ 


g 

and  the  point  of  application  of  the  resultant  force  may  be  con- 
ceived to  be  at  a  point  on  OG  at  a  distance  /  from  O  ;  and  this 
point  of  application  of  the  resultant  of  the  forces  which  pro- 
duce the  rotation  is  called  the  Centre  of  Percussion. 

If  p  =  radius  of  gyration  about  the  axis  through  (9,  and  if 
x0  :=  distance  from  (9  to  the  centre  of  gravity,  we  have 

XoSw 

Hence 


or,  in  words,  — 

The  radius  of  gyration  is  a  mean  proportional  between  the 
distance  1,  and  the  distance  x0,  betiveen  the  axis  of  oscillation  and 
the  centre  of  gravity. 

The  centre  of  percussion  with  respect  to  a  given  axis  of 
oscillation  O  has  been  defined  as  the  point  of  application  of  the 


122  APPLIED   MECHANICS. 

resultant  of  the  forces  which  cause  the  body  to  rotate  around  the. 
point  O. 

Another  definition  often  given  is,  that  it  is  the  point  at  which) 
if  a  force  be  applied,  there  will  be  no  shock  on  the  axis  of  oscilla- 
tion ;  and  these  two  definitions  are  equivalent  to  each  other. 

Let  the  particles  of  the  body  under  consideration  be  con- 
ceived, for  the  sake  of  simplicity,  to  be  distributed  along  a  single 
line  AB,  and  suppose  a  force  F  applied  at  D 
(Fig.  68).  Conceive  two  equal  and  opposite 
forces,  each  equal  to  F,  applied  at  C,  the  cen- 
tre of  gravity  of  the  body. 

\ Then  these  three  forces  are  equivalent  to 

a  single  force  ^ applied  at  the  centre  of  grav- 
ity C,  which  produces  translation  of  the  whole 
body ;  and,  secondly,  a  couple  whose  moment 
is  F(CD),  whose  effect  is  to  produce  rotation 
FIG.  68.  around  an  axis  passing  through  the  centre  of 

gravity  C.  Under  this  condition  of  things  it  is  evident  that  the 
centre  of  gravity  C  will  have  imparted  to  it  in  a  unit  of  time  a 

forward  velocity  equal  to  — -,  where  M  is  the  entire  mass  of  the 

body  ;  the  point  D  will  have  imparted  to  it  a  greater  forward 
velocity ;  while  those  points  on  the  upper  side  of  C  will  have 
imparted  to  them  a  less  and  less  velocity  as  they  recede  from 
C,  until,  if  the  rod  is  sufficiently  long,  the  particle  at  A  will 
acquire  a  backward  velocity. 

Hence  there  must  be  some  point  which  for  the  instant  in 
question  is  at  rest;  i.e.,  where  the  velocity  due  to  rotation  is  just 
equal  and  opposite  to  that  due  to  the  translation,  or  about 
which,  for  the  instant,  the  body  is  rotating :  and  if  this  point 
were  fixed  by  a  pivot,  there  would  be  no  stress  on  the  pivot 
caused  by  the  force  applied  at  D. 

An  axis  through  this  point  is  called  the  Instantaneous 
Axis. 


IMPACT   OR   COLLISION.  12$ 

§  102.  Interchangeability  of  the  Centre  of  Percussion 
and  Axis  of  Oscillation.  —  If  we  take,  as  axis  of  oscillation,  a 
line  perpendicular  to  the  plane  of  the  paper,  and  passing 
through  D,  then  will  O  be  the  new  centre  of  percussion. 

PROOF.  —  We  have  seen  (§  101)  that 


where  /  =  OD,  XQ  =  OC,  and  p  =  radius  of  gyration  about  an 
axis  through  O  perpendicular  to  the  plane  of  the  paper. 

Moreover,  if  /o0  represent  the  radius  of  gyration  about  an 
axis  through  C  perpendicular  to  the  plane  of  the  paper,  we  shall 
have 

P*  =  p02  +  x<* 


XQ 

Now  if  D  is  taken  as  axis  of  oscillation,  we  shall  have  for  the 
distance  lt  to  the  corresponding  centre  of  percussion, 


CD       I-  Xo' 

where  pI  =  radius   of   gyration  about  the   axis   of  oscillation 
through  D. 

/     .    Pi2       po2  +  CD2        p02        r  (j     ~\       j 

••  I'-CD  =    -CD~    "£D+CJ>-**+ «-*•)-*• 

Hence  the  new  centre  of  percussion  is  at  <9.     Q.  E.  D. 

§  103.  Impact  or  Collision.  —  Impact  or  collision  is  a 
pressure  of  inappreciably  short  duration  between  two  bodies. 

The  direction  of  the  force  of  impact  is  along  the  straight  line 
drawn  normal  to  the  surfaces  of  the  colliding  bodies  at  their 
point  of  contact,  and  we  may  call  this  line  the  line  of  impact. 


124  APPLIED   MECHANICS. 

The  action  that  occurs  in  the  case  of  collision  may  be  de- 
scribed as  follows  :  at  first  the  bodies  undergo  compression  ; 
the  mutual  pressure  between  them  constantly  increasing,  until, 
when  it  has  reached  its  maximum,  the  elasticity  of  the  mate- 
rials begins  to  overpower  the  compressive  force,  and  restore 
the  bodies  wholly  or  partially  to  their  original  shape  and  dimen- 
sions. 

Central  impact  occurs  when  the  line  joining  the  centres  of 
gravity  of  the  bodies  coincides  with  the  line  of  impact. 

Eccentric  impact  occurs  when  these  lines  do  not  coincide. 

Direct  impact  occurs  when  the  line  along  which  the  relative 
motion  of  the  bodies  takes  place,  coincides  with  the  line  of 
impact. 

Oblique  impact  occurs  when  these  lines  do  not  coincide. 

CENTRAL    IMPACT. 

§  104.  Equality  of  Action  and  Re-action.  —  One  funda- 
mental principle  that  holds  in  all  cases  of  central  impact  is  the 
equality  of  action  and  re-action  ;  in  other  words,  we  must  have, 
that,  at  every  instant  of  the  time  during  which  the  impact  is 
taking  place,  the  pressure  that  one  body  exerts  upon  the  other 
is  equal  and  opposite  to  that  exerted  by  the  second  upon  the 
first. 

The  direct  consequence  of  this  principle  is,  that  the  algebraic 
sum  of  the  momenta  of  the  two  bodies  before  impact  remains 
unaltered  by  the  impact,  and  hence  that  this  sum  is  just  the 
same  at  every  instant  of,  and  after,  the  impact. 

If  we  let 

mlt  m2,  be  the  respective  masses, 
clt    c2,  their  respective  velocities  before  impact, 

vu   v2,  their  respective  velocities  after  impact, 

i/,   v" ,  their  respective  velocities  at  any  given  instant  during 
the  time  while  impact  is  taking  place, 


CO-EFFICIEArT  OF  RESTITUTION.  125 

then  we  must  have  the  following  two  equations  true ;  viz.,  — 
m1vl  +  m2v2  =  mlcl  +  m2c2,  (i) 

mjf  4-  m2v"  =  mlcl  4-  m2c2.  (2) 

§  105.   Velocity  at  Time  of  Greatest  Compression.  —  At 

the  instant  when  the  compression  is  greatest  —  i.e.,  at  the 
instant  when  the  elasticity  of  the  bodies  begins  to  overcome 
the  deformation  due  to  the  impact,  and  to  tend  to  restore  them 
to  their  original  forms  —  the  values  of  v'  and  v"  must  be  equal 
to  each  other;  in  other  words,  the  colliding  bodies  must  be 
moving  with  a  common  velocity 

v  =  v'  =  v".  (i) 

To  determine  this  velocity,  we  have,  from  equation  (2),  §  104, 
combined  with  (i), 

v  =  m^  +  m*c\  (2) 

ml  4-  m2 

§  106.  Co-efficient  of  Restitution.  —  In  order  to  determine 
the  values  vlt  v2,  of  the  velocities  after  impact,  we  need  two 
equations,  and  hence  two  conditions.  One  of  them  is  fur- 
nished by  equation  (i),  §  104.  The  second  depends  upon  the 
nature  of  the  material  of  the  colliding  bodies,  and  we  may  dis- 
tinguish three  cases  :  — 

i°.  Inelastic  Impact.  —  In  this  case  the  velocity  lost  up  to 
the  time  of  greatest  compression  is  not  regained  at  all,  and 
the  velocity  after  impact  is  the  common  velocity  ^  at  the  instant 
of  greatest  compression.  In  this  case  the  whole  of  the  work 
used  up  in  compressing  the  bodies  is  lost,  as  none  of  it  is 
restored  by  the  elasticity  of  the  material. 

2°.  Elastic  Impact.  —  In  this  case  the  velocity  regained 
after  the  greatest  compression,  is  equal  and  opposite  to  that 
lost  up  to  the  time  of  greatest  compression  ;  therefore 

v  —  z/j  =  cv  —  v.        (i)  v2  —  v  —  v  —  c2.     (2) 


126  APPLIED   MECHANICS. 

We  may  also  define  this  case  as  that  in  which  the  work  lost 
in  compressing  the  bodies  is  entirely  restored  by  the  elasticity 
of  the  material,  so  that 


j     .       z2  , 

--  --  —  --  r 

2222 

Either  condition  will  lead  to  the  same  result. 

3°.  Imperfectly  Elastic  Impact.  —  In  this  case  a  part  only 
of  the  velocity  lost  up  to  the  time  of  greatest  compression  is 
regained  after  that  time. 

If,  when  the  two  bodies  are  of  the  same  material,  we  call  e 
the  co-efficient  of  restitution,  then  we  shall  so  define  it  that 

v  —  vl 


c*  —  v       v  —  c2 

or,  in  words,  the  co-efficient  of  restitution  is  the  ratio  of  the 
velocity  regained  after  compression  to  that  lost  previous  to 
that  time. 

In  this  case  only  a  part  of  the  work  done  in  producing  the 
compression  is  regained,  hence  there  is  loss  of  energy.  Its 
amount  will  be  determined  later. 

Strictly  speaking,  all  bodies  belong  to  the  third  class  ;  the 
value  of  e  being  always  a  proper  fraction,  and  never  reaching 
unity,  the  value  corresponding  to  perfect  elasticity  ;  nor  zero, 
the  value  corresponding  to  entire  lack  of  elasticity. 

§  107.  Inelastic  Impact.  —  In  this  case  the  velocity  after 
impact  is  the  common  velocity  at  the  time  of  greatest  com- 
pression ;  hence 

v  =  v,  =  v2  (i) 


(2) 


And  for  the  loss  of  energy  due  to  impact  we  have 


m2c2        ,  ^v* 

1  ---  (M!  -f-  m2)  —  , 

2  2 


ELASTIC  IMPACT.  12  J 


which,  on  substituting  the  value  of  v,  reduces  to 

,,"'?„)  ('•  -  <•>••  (3) 

2(m1  •+-  m2) 

§  1 08.  Elastic  Impact.  —  In  this  case  we  have,  of  course, 
the  condition,  equation  (i),  §  104, 


m\v-L  +  ^2^2  ==  m^c-i  -f* 
and,  for  second  equation,  we  may  use  equation  (3),  §  106  ;  viz., 

w^!2    ,    m2v22  _  mj?        m^c^ 

2222 

Combining  these  two  equations,  we  shall  obtain 


ml 


We  can  obtain  the  same  result  without  having  to  solve  an 
equation  of  the  second  degree,  by  using  instead  the  equations 
(i)  and  (2)  of  §  106,  together  with  (i)  of  §  104;  i.e.,  — 

mlvI  -f-  mzvz  =  m1fl  +  m^\ 


or 

and  (§  105) 


l  ~\~ 


As  the  result  of  combining  these  equations,  and  eliminating 
v,  we  should  obtain  equations  (i)  and  (2),  as  above,  for  the  values 
of  z\  and  v2.  In  this  case  the  energy  lost  by  the  collision  is 
zero. 


128  APPLIED   MECHANICS. 

§  109.   Special  Cases  of  Inelastic  Impact.  —  (a)   Let  the 
mass  m2  be  at  rest.     Then  c2  —  o, 


v  =      m'c' 


<«     +     ™* 

.' .     Loss  of  energy  =      m*m*     ££.  (2) 

(£)  Let  w2  be  at  rest,  and  let  m2  =  oo  ;  i.e.,  let  the  mass  ;;/r 
strike  against  another  which  is  at  rest,  and  whose  mass  is  in- 
finite. We  have 

m2  =  oo ,        c2  =  o, 

-  =  o,  (3) 


m 


W.  ^r  Wi^r 

Loss  of  energy  =  -  -  --  -  =  —±-L,  (4) 


or  the  moving  body  is  reduced  to  rest  by  the  collision,  and  all 
its  energy  is  expended  in  compression. 

(c)  Let  mlcl  =  —m2c2  ;  i.e.,  let  the  two  bodies  move  towards 
each  other  with  equal  momenta  : 


o,  (5) 


and  the  loss  of  energy  =  ^^  -f  ^£2!,  (6) 

2  2 

the  entire  energy  being  lost. 

§110.   Special    Cases    of   Elastic    Impact.  —  (a)  Let  the 
mass  m2  be  at  rest.     Then  c2  =.  o, 


EXAMPLES   OF  ELASTIC  AND   INELASTIC   IMPACT.       I2g 

(b)  Let  m2  be  at  rest,   and  let  m2  —  oo .     Then  we  have 

'2     =0, 

^+     I 

m2 
V2  =  o.  (4) 

Hence  the  moving  body  retraces  its  path  in  the  opposite  direc- 
tion with  the  same  velocity. 

(c)  Let  m^,  =  —m2c2.     Then   our  equations   of   condition 

become 

WiVi  +  m2v2  =  o, 

2222 

and  from  these  we  readily  obtain 


i.e.,  both  bodies  return  on  their  path  with  the  same  velocity 
with  which  they  approached  each  other. 

§in.   Examples  of  Elastic  and  of  Inelastic  Impact. 

1.  With  what  velocity  must  a  body  weighing  8  pounds  strike  one 
weighing  25  pounds  in  order  to  communicate  to  it  a  velocity  of  2  feet 
per  second,  (a)  when  the  bodies  are  perfectly  elastic,  (b)  when  wholly 
inelastic. 

2.  Suppose  sixteen  impacts  per  minute  take  place  between  two  bodies 
whose  weights  are  respectively  1000  and  1200  pounds,  their  initial  velo- 
cities being  5  and  2  feet  per  second  respectively  :  find  the  loss  of  energy, 
the  bodies  being  inelastic. 

§  112.  Imperfect   Elasticity.  —  In  this  case  we  have  the 

relations  (see  §  106) 

V  —  Vi  _ 

v  -  c2 


130  APPLIED   MECHANICS. 

where 

v  =  m£  +  ™*f*; 

and  we  have  also 

m^,  -h  m2v2  =  m^  -f  m2c2. 

Determining  from  them  the  values  of  z/x  and  vz,  we  obtain 
Vs  =  »(i  +  e)  -  ect,  (i) 

v2  =  »(i  +  0  -  ecv  (2) 

or,  by  substituting  for  v  its  value, 


These  may  otherwise  be  put  in  the  form 

--  -'«-(«+•)  «•  -  O,       (5) 


Moreover,  we  have  for  the  loss  of  energy  due  to  impact 


E  =       (^2  -  ^2)  +  ^(r22  -  v2 

2  2 


or 


but,  from  (5)  and  (6)  respectively, 

fs   -   Vl   =    (T    +   <?)  ^2(^*1 

Wj  -f  w2 
^  -  «,a  =  _  (i  +  g)«,( 

W,    -f- 


IMPERFECT  ELASTICITY.  131 


2(ml 
/.     E  =      .  m 


But,  from  (i)  and  (2), 


or 


When  ^  =  i,  or  the  elasticity  is  perfect,  this  loss  of  energy 
becomes  zero. 

When  e  =  o,  or  the  bodies  are  totally  inelastic,  then  the  loss 
of  energy  becomes 


/  L       »**    \      v  mf    *  V     / 

*\rr*l    ~T~    "*2/ 

as  has  been  already  shown  in  §  107. 

An  interesting  fact  in  this  connection  is,  that  since  (8)  is 
the  work  expended  in  producing  compression,  and  (7)  is  the 
work  lost  in  all,  therefore  the  work  restored  by  the  elasticity  of 
the  body  is 


so  that  e2,  or  the  square  of  the  co-efficient  of  restitution,  is  the 
ratio  of  the  work  restored  by  the  elasticity  of  the  bodies,  to 
the  work  expended  in  compressing  the  bodies  up  to  the  time 
of  greatest  compression. 


132  APPLIED   MECHANICS. 

§113.    Special  Cases.  —  (a)    Let  m2  be  at  rest,  therefore 
2  =  o.     Then  we  shall  have 

(     _^(i  +  *)  1  =,  ,*.-**.          (I) 

(  #*i  +  *«2   )  mt  +  m2 


and  for  loss  of  energy 


When  #z2  =  oo  ,  and  £2  =  o,  we  have 

PI  =  —  ^i,  (4) 

^2  =      o, 

^  =  (I  _  ,2)  ^!.  (5) 

When  mlcl  =  —m2c2,  then 


m,)  ^ 


§  1  14.    Values   of    e   as    Determined   by   Experiment. 
Since  we  have 


_ 
&  — 


IMPERFECT  ELASTICITY.  133 

we  shall  have,  when 

m2  =  oo      and      c2  =  o, 


m  x  4- 
Hence 


Now,  if  we  let  a  round  ball  fall  vertically  upon  a  horizontal 
slab  from  the  height  Hy  we  shall  have  for  the  velocity  of  ap- 
proach 


and  if  we  measure  the  height  h  to  which  it  rises  on  its  rebound, 
we  shall  have 


Hence 

In  this  way  the  value  of  e  can  be  determined  experimentally 
for  different  substances. 

Newton  found  for  values  of  e:  for  glass,  ||;  for  steel,  |~ 
and  Coriolis  gives  for  ivory  from  0.5  to  0.6. 

On  the  other  hand,  if  we  desired  to  adopt  as  our  constant 
the  ratio  of  the  work  restored,  to  the  work  spent  in  compres- 
sion, we  should  have  for  our  constant  ^2,  and  hence  the  squares 
of  the  preceding  numbers. 

EXAMPLES. 

i.  If  two  trains  of  cars,  weighing  120000  and  160000  Ibs.,  come 
into  collision  when  they  are  moving  in  opposite  directions  with  veloci- 
ties 20  and  15  feet  per  second  respectively,  what  is  the  loss  of  mechan- 
ical effect  expended  in  destroying  the  locomotives  and  cars  ? 


134  APPLIED  MECHANICS. 

2.  Two  perfectly  inelastic   balls   approach   each   other  with   equal 
velocities,  and  are  reduced  to  rest  by  the  collision ;  what  must  be  the 
ratio  of  their  weights  ? 

3.  Two  steel  balls,  weighing  10  Ibs.  each,  are  moving  with  velocities 
5  and  10  feet  per  second  respectively,  and  in  the  same  direction  :  find 
their  velocities  after  impact,  the  fastest  ball  being  in  the  rear,  and  over- 
taking the  other ;  also  the  loss  of  mechanical  effect  due  to  the  impact, 
assuming  e  =  0.55. 

§  115.  Oblique  Impact. 

Let  mlt  m2,  be  the  masses  of  the  colliding  bodies ; 
clt  c2,  their  respective  velocities  before  impact ; 
an  a2,  the  angles  made  by  clt  c2,  with  the  line  of  centres ; 
vu  v2y  the  components  of  the  velocities  after  impact ; 
£,  cos  a,,  c2  cos  a2,  the  components  of  clt  c2,  along  the  line  of 

centres ; 
cl  sin  a,,  c2  sin  a2,  the  components  of  cu  c2t  at  right  angles  to 

the  line  of  centres  ; 
v  the  common  component  of  the  velocity  at  the  instant  of 

greatest  compression  along  line  of  centres ; 
i/t  v",  actual  velocities  after  impact ; 
a',  a",  angles  they  make  with  line  of  centres ; 
vj,  v",  actual  velocities  when  compression  is  greatest ; 
a/,  a/',  angles  they  make  with  line  of  centres. 

Then  we  shall  have,  by  proceeding  in  the  same  way  as  was  done 
m  §  112, 

V,  =  ^COSa,  —   (l  +  e) ~ (VjCOSa,  —  ^2COSa2),     (l) 

#*i  H-  ^2 

V2  =  C2  COS  a2  +   (l 


OBLIQUE  IMPACT.  135 


(3) 
(4) 

(5) 
(6) 

(7) 

(8) 
(9) 

(10) 

tf  =  VW  +  ^sin'a,, 

if'  =  y^22  +  ^22  sin2  a,, 
COS  a'  =  ^, 

COS  a"  =  J&, 
V 

Vc  =  Vz'2  +  <Tj2  sin2  ax, 

v/'  =  ^v2  H~  ^sin2^, 

00.*'  -J, 

COS  a/'  =  -^ 


/?* 


And  for  the  energy  lost  in  impact,  we  have 


,        _  . 

2(ml  -f-  w2) 

When  the  bodies  are  perfectly  elastic, 

g  =  i, 
and  equations  (i),  (2),  and  (12)  become  respectively 

Vi  =  ^  COS  a,  --  2  -  (fi  cos  af  _  ^  cos  ^ 


C2  COS  a2  H  --  ?^±  —  (fs  COS  a,  —  ^2  COSa,), 

ml  +  m2 


The  rest  remain  the  same  in  form. 

When  the  bodies  are  totally  inelastic, 

*    SB     O, 


136  APPLIED   MECHANICS. 

and  equations  (i),  (2),  and  (12)  become  respectively 

#,   =   fj  COS  a,    —    -  -  -  (^TjCOStt!    —   ^  COS  03), 

m^  +  mz 

V2  =  c2  cos  a2  H  --  -  -  (Cj.  cos  ax  —  <r2  cos  03), 
/«,  +  w2 

,  -  r2cosa2)2. 


2(ml  4-  m2) 
The  rest  remain  the  same  in  form. 

§  116.    Impact   of   Revolving   Bodies.  —  Let  the  bodies  A 
and  B  revolve  about  parallel  axes,  and  impinge  upon  each  other. 
Draw  a  common   normal  at  the  point  of  contact.      This 
common  normal  will  be  the  line  of  impact. 
Let  c,  —  angular  velocity  of  A  before  impact, 
c2  =  angular  velocity  of  B  before  impact, 
o>x  =  angular  velocity  of  A  after  impact, 
<o2  =  angular  velocity  of  B  after  impact, 
^  —  perpendicular  from  axis  of  A  on  line  of  impact, 
az  =  perpendicular  from  axis  of  B  on  line  of  impact, 
7X  =  moment  of  inertia  of  A  about  its  axis, 
72  =  moment  of  inertia  of  B  about  its  axis. 

Then  we  shall  have 

alel  =  Ci  =  linear  velocity  of  A  at  point  of  contact  before  impact  ; 

a2€2  =  c2  =  linear  velocity  of  B  at  point  of  contact  before  impact  ; 

al(j}l  =  Vj.  =  linear  velocity  of  A  at  point  of  contact  after  impact  ; 

a2<a2  —  v2  =  linear  velocity  of^  at  point  of  contact  after  impact; 


2g 


//  V  2 

_  /  _!_  \JL  =  actual  energy  of  A  before  impact  : 

\*i2hg 

/  /    \C  2 

=  (  —  ^  ]-L-  —  actual  energy  of  B  before  impact  ; 
\a2*)2& 

-^-  =  (  —  -  )—  =  actual  energy  of  A  after  impact  : 

2g  W/2£- 

^-  =  (  —  -  j-2-  =  actual  energy  of  B  after  impact  ; 
2S        \a22/  2g 


IMPACT  OF  REVOLVING   BODIES.  137 

Hence  it  follows  that  we  have  the  case  explained  in  §  112  for 
imperfectly  elastic  impact,  provided  only  we  write 


-—  instead  of  m^g     and     —  instead  of  m2g. 
a,2  a22 


-—  instead 
a,2 

Hence  we  shall  have 

o>,  =  e,   -   *,(«,«,    -   <*2e2)         8  f2  r     ,(l   +  ')»        (0 

/,#,  -f/x 

W2  =   C2    4-    tfaOi*!    —    ^2^2)7  -  '—  -  (l    +   <?),          (2) 

!&*     +     /I^22 

The  case  of  perfect  elasticity  is  obtained  by  making  e  =  i. 
The  case  of  total  lack  of  elasticity  is  obtained  by  making 

f  —  O. 

In  the  latter  case  the  loss  of  energy  is 

7r/2  x  v 


as  can  be  seen  by  substituting  the  proper  values  in  equation  (8), 
§112. 


138  APPLIED   MECHANICS. 


CHAPTER   III. 
ROOF-TRUSSES. 

§  117.  Definitions  and  Remarks. —  The  term  "truss"  may 
be  applied  to  any  framed  structure  intended  to  support  a  load. 

In  the  case  of  any  truss,  the  external  loads  may  be  applied 
only  at  the  joints,  or  some  of  the  truss  members  may  support 
loads  at  points  other  than  the  joints. 

In  the  latter  case  those  members  are  subjected,  not  merely 
to  direct  tension  or  compression,  but  also  to  a  bending-action, 
the  determination  of  which  we  shall  defer  until  we  have  studied 
the  mode  of  ascertaining  the  stresses  in  a  loaded  beam ;  and 
we  shall  at  present  confine  ourselves  to  the  consideration  of 
the  direct  stresses  of  tension  and  compression. 

For  this  purpose  any  loads  applied  between  two  adjacent 
joints  must  be  resolved  into  two  parallel  components  acting  at 
those  joints,  and  the  truss  is  then  to  be  considered  as  loaded  at 
the  joints.  By  this  means  we  shall  obtain  the  entire  stresses  in 
the  members  whenever  the  loads  are  concentrated  at  the  joints; 
and,  when  certain  members  are  loaded  at  other  points,  our  re- 
sults will  be  the  direct  tensions  and  compressions  of  these  mem- 
bers, leaving  the  stresses  due  to  bending  yet  to  be  determined. 

A  tie  is  a  member  suited  to  bear  only  tension. 

A  strut  is  a  member  suited  to  bear  compression. 

§  1 1 8.  Frames  of  Two  Bars.  —  Frames  of  two  bars  may 
consist,  (i)  of  two  ties  (Fig.  69),  (2)  of  two  struts  (Fig.  70), 
(3)  of  a  strut  and  a  tie  (Fig.  71). 


FRAMES   OF   TWO  BARS. 


139 


CASE   I.    Two    Ties   (Fig.  69).  —  Let   the   load   be   repre- 
sented graphically  by  CF  =  W.    a 
Then    if    we   resolve  it   into 
two  components,  CD  and  CE, 
acting  along  CB  and  CA  re- 
spectively, CD  will  represent 
graphically  the  pull  or  tension 
in  the  tie  CB,  and  CE  that  in 
the  tie  CA. 

The  force  acting  on  CB  at 
B  is  equal   and    opposite   to 


FIG.  69. 


CD,  while  that  acting  on  CA  at  A  is  equal  and  opposite  to  CE. 
To  compute  these  stresses  analytically,  we  have 


CE  =  CF 


sin  CFE 


=  W 


sin  2 


sin  CEF  sin(Y  +  /,)' 


CD  =  CF 


sin  CFD 
sin  CDF 


=  W 


sin/, 


sin(Y  +  /,) 

CASE  II.  Two  Struts  (Fig.  70).  —  Let  the  load  be  repre- 
sented graphically  by  CF=  W. 
Then  will  the  components  CD 
and  CE  represent  the  thrusts 
in  the  struts  CB  and  CA  re- 
spectively, and  the  re-actions 
of  the  supports  at  B  and  A 
will  be  equal  and  opposite  to 
them.  For  analytical  solution, 
we  derive  from  the  figure 


FIG.  70. 


CE  =  W 


smi. 


sin(*  -f- 


CD  =  W 


sin* 


sin(i  -f  *,) 


CASE  III.  A  Strut  and  a  Tie  (Fig.  71).  —  Let  the  load  be 
represented  graphically  by  CF  =  W.  Resolve  it,  as  before, 
into  components  along  the  members  of  the  truss.  Then  will 


140  APPLIED  MECHANICS. 

CE  represent  the  tension  in  the  tie  AC,  and  CD  will  represent 
the  thrust  in  the  strut  BC ;  and  we  may 
deduce  the  analytical  formulae  as  before. 

§  1 19.  Stability  for  Lateral  Deviations. 
-In  Case  I,  if  the  joint  C  be  moved  a  little 
out  of  the  plane  of  the  paper,  the  load  at 
C  has  such  a  direction  that  it  will  cause  the 
truss  to  rotate  around  AB  so  as  to  return  to 
its  former  position  ;  hence  such  a  frame  is 
stable  as  regards  lateral  deviations. 

In  Case  II   the  effect  of   the  load,  if  C 
were  moved  a  little  out  of  the  plane  of  the 
paper,  would  be  to  cause  rotation  in  such  a  way  as  to  overturn 
the  truss ;    hence  such  a  frame  is  unstable  as  regards  lateral 
deviations. 

In  Case  III  the  stability  for  lateral  deviations  will  depend 
upon  whether  the  load  CF  =  W  is  parallel  to  AB,  is  directed 
away  from  it  or  towards  it.  If  the  first  is  the  case  (i.e.,  if  A  is 
the  point  of  suspension  of  the  tie),  the  frame  is  neutral,  as  the 
load  has  no  effect,  either  to  restore  the  truss  to  its  former  posi- 
tion, or  to  overturn  it ;  if  the  second  is  the  case  (i.e.,  if  At  is 
the  point  of  suspension  of  the  tie),  the  truss  is  stable ;  and,  if 
the  third  is  the  case  (i.e.,  if  A±  is  the  point  of  suspension  of  the 
tie),  it  is  unstable  as  regards  lateral  deviations. 

§  1 20.   General  Methods  for  Determining  the  Stresses  in 
Trusses.  —  In  the  determination  of  the  stresses  as  above,  it 
would  have  been  sufficient  to  construct  only  the  triangle  CFD 
by  laying  off  CF=  W  to  scale,  and  then  drawing  CD  parallel 
to  CB,  and  FD  parallel  to  CA,  and  the  triangle  CFD  would  have 
given  us  the  complete  solution  of  the  problem.     Moreover,  the 
determination  of  the  supporting  forces  of  any  truss,  and  of  the 
stresses  in  the  several  members,  is  a  question  of  equilibrium. 
Adopting  the  following  as  definitions,  viz.,  — 
External  forces  are  the  loads  and  supporting  forces, 


TRIANGULAR  FRAME.  14! 

Internal  forces  are  the  stresses  in  the  members  : 
we  must  have 

i°.  The  external  forces  must  form  a  balanced  system;  i.e., 
the  supporting  forces  must  balance  the  loads. 

2°.  The  forces  (external  and  internal)  acting  at  each  joint 
of  the  truss  must  form  a  balanced  system  ;  i.e.,  the  external 
forces  (if  any)  at  the  joint  must  be  balanced  by  the  stresses  in 
the  members  which  meet  at  that  joint. 

3°.  If  any  section  be  made,  dividing  the  truss  into  two  parts, 
the  external  forces  which  act  upon  that  part  which  lies  on  one 
side  of  the  section,  must  be  balanced  by  the  forces  (internal) 
exerted  by  that  part  of  the  truss  which  lies  on  the  other  side 
of  the  section,  upon  the  first  part. 

The  above  three  principles,  the  triangle,  and  polygon  of 
forces,  and  the  conditions  of  equilibrium  for  forces  in  a  plane, 
enable  us  to  determine  the  stresses  in  the  different  members 
of  roof  and  bridge  trusses. 

§121.  Triangular  Frame.  —  Given  the  triangular  frame 
ABC  (Fig.  72),  and  given  the  load  W  at  C  in  magnitude  and 
direction,  given  also  the  N 
direction  of  the  support- 
ing force  at  B,  to  find  the 
magnitude  of  this  support- 
ing force,  the  magnitude 
and  direction  of  the  other 
supporting  force,  and  the 
stresses  in  the  members. 

SOLUTION. — Join  A  \  b 

with  D,  the  point  of  inter-  FIG.  72. 

section  of  the  line  of  direction  of  the  load  and  the  line  BE. 
Then  will  DA  be  the  direction  of  the  other  supporting  force  ; 
for  the  three  external  forces,  in  order  to  form  a  balanced  sys- 
tem, must  meet  in  a  point,  except  when  they  are  parallel. 
Then  draw  ab  to  scale,  parallel  to  CD  and  equal  to  W.  From 


142 


APPLIED  MECHANICS. 


a  draw  ac  parallel  to  BD,  and  from  b  draw  be  parallel  to  AD ; 
then  will  the  triangle  abca  be  the  triangle  of  external  forces, 
the  sides  ab,  be,  and  ca,  taken  in  order,  representing  respectively 
the  load  W,  the  supporting  force  at  A,  and  the  supporting  force 
at  A 

Then  from  a  draw  ad  parallel  to  BC,  and  from  c  draw  cd 
parallel  to  AB ;  then  will  the  triangle  acd  be  the  triangle  of 
forces  for  the  joint  B,  and  the  sides  ca,  ad,  and  dc,  taken  in 
order,  will  represent  respectively  the  supporting  force  at  B,  the 
force  exerted  by  the  bar  BC  at  the  point  B,  and  the  force 
exerted  by  the  bar  AB  at  the  point  B. 

Since,  therefore,  the  force  ad  exerted  by  the  bar  CB  at  B 
is  directed  away  from  the  bar,  it  follows  that  CB  is  in  compres- 
sion ;  and,  since  the  force  dc  exerted  by  the  bar  AB  at  B  is 
directed  towards  the  bar,  it  follows  that  AB  is  in  tension. 

In  the  same  way  bdc  is  the  triangle  of  forces  for  the  point 
A ;  the  sides  be,  cd,  and  db  representing  respectively  the  sup- 

porting  force  at  A,  the  force 
exerted  by  the  bar  AB  at  A, 
and  the  force  exerted  by  the 
bar  AC  at  A. 

The  bar  AB  is  again  seen  to 
be  in  tension,  as  the  force  cd 
exerted  by  the  bar  AB  at  A  is 
directed  towards  the  bar. 

So  likewise  the  triangle  abd 
is  the  triangle  of  forces  for  the 
point  C. 

Fig.  73  shows  the  case  when 
the  supporting  forces  meet  the  load-line  above,  instead  of 
below,  the  truss. 

§  122.  Triangular  Frame  with  Load  and  Supporting 
Forces  Vertical.  —  Fig.  74  shows  the  construction  when  the 
load  and  also  the  supporting  forces  are  vertical.  In  this  case 


\ 


FIG.  73. 


BOW'S  NOTATION. 


143 


FIG.  74. 


the  diagram  becomes  very  much  simplified,  the  triangle  of 
external  forces  abd  becom- 
ing a  straight  line.  The 
diagram  is  otherwise  con- 
structed just  like  the  last 
one. 

§  123.   Bow's  Notation. 
—  The  notation  devised  by 
Robert  H.  Bow  very  much 
simplifies   the   construction   of    the   stress   diagrams   of   roof- 
trusses. 

This  notation  is  as  follows  :  Let  the  radiating  lines  (Fig.  75) 
represent  the  lines  of  action  of  a  system  of  forces  in  equilib- 
rium, and  let  the  polygon  abcdefa  be  the  polygon  representing 

these  forces  in  magnitude 
and  direction  ;  then  denote 
the  sides  of  the  polygon 
in  the  ordinary  way,  by 
placing  small  letters  at  the 
vertices,  but  denote  the 
radiating  lines  by  capital 
letters  placed  in  the  angles. 
Thus  the  line  AB  is  the 
line  of  direction  of  the 
force  aby  etc.  In  applying  the  notation  to  roof-trusses,  we  letter 
the  truss  with  capital  letters  in  the  spaces,  and  the  stress  dia- 
gram with  small  letters  at  the  vertices.  If,  then,  in  drawing 
the  polygon  of  equilibrium  for  any  one  joint  of  the  truss,  we 
take  the  forces  always  in  the  same  order,  proceeding  always 
in  right-handed  or  always  in  left-handed  rotation,  we  shall  be 
led  to  the  simplest  diagrams.  Hereafter  this  notation  will  be 
used  exclusively  in  determining  the  stresses  in  roof-trusses. 

§124.  Isosceles  Triangular  Frame:  Concentrated  Load 
(Fig.  76.)  —  Let  the  load  W  act  at  the  apex,  the  supporting 


FIG.  75. 


144 


APPLIED  MECHANICS. 


FIG.  76. 


forces  being  vertical ;   each  will  be  equal  to  \  W :   hence  the 
polygon  of  external  forces  will  be  the  triangle  abc,  the  sides  of 

which,  ab,  be,  and  ca,  all  lie  in 
one  straight  line.  Then  begin 
at  the  left-hand  support,  and 
proceed  again  in  right-handed 
rotation,  and  we  have  as  the  tri- 
angle of  forces  at  this  joint  cad, 
the  forces  ca,  ad,  and  dc,  these 
being  respectively  the  support- 
ing force,  the  stress  in  AD,  and 
that  in  DC ;  the  directions  of 
these  forces  being  indicated  by 
the  order  in  which  the  letters  follow  each  other :  thus,  ca  is  an 
upward  force,  ad  is  a  downward  force  ;  and,  this  being  the 
force  exerted  by  the  bar  AD  at  the  left-hand  support,  we  con- 
clude that  the  bar  AD  is  in  compression.  Again  :  dc  is 
directed  towards  the  right,  or  towards  the  bar  itself,  and  hence 
the  bar  DC  is  in  tension.  The  triangle  of  forces  for  the  other 
support  is  bed,  and  that  for  the  apex  abd. 

§  125.  Isosceles  Triangular  Frame:  Distributed  Load. — 
Let  the  load  W  be  uniformly  distributed  over  the  two  rafters 
AF  and  FB  (Fig.  77)  ;  then  will 
these  two  rafters  be  subjected  to 
a  direct  stress,  and  also  to  a  bend- 
ing action  :  and  if  we  resolve  the 
load  on  each  rafter  into  two  com- 
ponents at  the  ends  of  the  rafter, 
then,  considering  these  components 
as  the  loads  at  the  joints,  we  shall 
determine  correctly  by  our  diagram  the  direct  stresses  in  all 
the  bars  of  the  truss. 

The  load  distributed  over  AF  is  — ;  and  of  this,  one-half  is 


FIG.  77. 


POLYGONAL   FRAME. 


the  component  at  the  support,  and  one-half  at  the  apex,  and 

similarly  for  the  other  rafter.     This  gives  as  our  loads,  —  at 

4 

each  support,  and--  at  the  apex.  The  polygon  of  external 
forces  is  eabcde,  where  the  sides  are  as  follows :  — 

W  W     ,         W        ,       W      ,        W 

ea  =  — ,     ab  —  — ,     be  —  — ,     cd  =  — ,     de  =  — . 
42422 

Then,  beginning  at  the  left-hand  support,  we  shall  have  for  the 
polygon  of  forces  the  quadrilateral  deafd,  where  de  =  —  =  sup- 
porting force,  ea  = --  =  downward  load  at  support,  af  — 

4 

stress  in  AF  (compression),  fd  —  stress  in  FD  (tension).  The 
polygon  for  the  apex  is  abf,  and  that  for  the  right-hand  support 
cdfbc. 

§  1 26.   Polygonal  Frame Given  a  polygonal  frame  (Fig. 

78)  formed  of  bars  jointed  together  at  the  vertices  of  the  angles, 
and  free  to  turn  on  these  joints, 
it  is  evident,  that,  in  order  that 
the  frame  may  retain  its  form, 
it  is  necessary  that  the  direc- 
tions of,  and  the  proportions 
between,  the  loads  at  the  dif- 
ferent joints,  should  be  speci- 
ally adapted  to  the  given  form  : 
otherwise  the  frame  will  change 
its  form.  We  will  proceed  to 
solve  the  following  problem  : 
Given  the  form  of  the  frame, 
the  magnitude  of  one  load  as  AB,  and  the  direction  of  all  the 
external  forces  (loads  and  supporting  forces)  except  one,  we 
shall  have"  sufficient  data  to  determine  the  magnitudes  of  all, 


\ 
\ 

\ 
\ 

F 

FIG.  78. 


146 


APPLIED   MECHANICS. 


and  the  direction  of  the  remaining  external  forces,  and  also  the 
stresses  in  the  bars 

Let  the  direction  of  all  the  loads  be  given,  and  also  that  of 
the  supporting  force  EF,  that  of  the  supporting  force  AF  being 
thus  far  unknown  ;  and  let  the  magnitude  of  AB  be  given. 
Then,  beginning  at  the  joint  ABG,  we  have  for  triangle  of 
forces  abg  formed  by  drawing  ab  ||  and  =  AB,  then  drawing 
ga  ||  AG,  and  bg  ||  BG ;  ga  and  bg  both  being  thrusts.  Then, 
passing  to  the  joint  BCG,  we  have  the  thrust  in  BG  already 
determined,  and  it  will  in  this  case  be  represented  by  gb.  If, 
now,  we  draw  be  ||  BC,  and  gc  ||  GC,  we  shall  have  determined 
the  load  BC  as  be,  and  we  shall  have  eg  and  gb  as  the  thrusts 
in  CG  and  GB  respectively.  Continuing  in  the  same  way,  we 
obtain  the  triangles  gcd,  gde,  and  gfe,  thus  determining  the 
magnitudes  of  the  loads  cd,  de,  and  of  the  supporting  force  ef; 
and  then  the  triangle  gaf,  formed  by  joining  a  and/,  gives  us  af 
for  the  magnitude  and  direction  of  the  left-hand  support.  The 
polygon  abcdefa  of  external  forces  is  called  the  Force  Polygon, 
while  the  frame  itself  is  called  the  Equilibrium  Polygon. 

§  127.    Polygonal    Frame    with    Loads    and    Supporting 

Forces  Vertical In  this  case  (Fig.   79)  we  may  give  the 

form  of  the  frame  and  the  mag- 
nitude of  one  of  the  loads,  to 
determine  the  other  loads  and 
the  supporting  forces,  and  also 
the  stresses  in  the  bars ;  or  we 
may  give  the  form  of  the  frame 
and  the  magnitude  of  the  re- 
sultant of  the  loads,  to  find  the 
loads  and  supporting  forces.  In 
the  former  case  let  the  load  AB » 
be  given.  Then,  proceeding  in 
the  same  way  as  before,  we  find  the  diagram  of  Fig.  79 ;  the 
polygon  of  external  forces  abcdefa  falling  all  in  one  straight  line. 


FIG.  79. 


FUNICULAR  POLYGON. —  TRIANGULAR    TRUSS. 


147 


If,  on  the  other  hand,  the  whole  load  ae  be  given,  we  observe 
that  this  is  borne  by  the  stresses  in  the  extreme  bars  AG  and 
GE ;  hence,  drawing  ag  ||  AG,  and  eg  ||  EG,  we  find  eg  and  ga 
as  the  stresses  in  EG  and  GA  respectively.  Then,  proceeding 
to  the  joint  ABG,  we  find,  since ' ga  is  the  force  exerted  by 
GA  at  this  point,  that,  drawing  gb  ||  GB,  we  shall  have  ab  as 
the  part  of  the  load  acting  at  the  joint  ABG,  etc. 

§  128.  Funicular  Polygon.  —  If  the  frame  of  Fig.  79  be 
inverted,  we  shall  have  the 
case  of  Fig.  80,  where  all 
the  bars,  except  FG,  are  sub- 
jected to  tension;  FG  itself 
being  subjected  to  compres- 
sion. The  construction  of  the 
diagram  of  stresses  being  en-  \ 
tirely  similar  to  that  already 
explained  for  Fig.  79,  the  ex- 
planation will  not  be  repeated 
here.  If  the  compression 
piece  be  omitted,  the  case 
becomes  that  of  a  chain  hung 
at  the  upper  joints  (the  supporting  forces  then  becoming  iden- 
tical with  the  tensions  in  the  two  extreme  bars),  the  line  gf 
would  then  be  omitted  from  the  diagram,  and  the  polygon  of 
external  forces  would  become  abcdega. 

§  129.  Triangular  Truss  :  Wind  Pressure.  —  Inasmuch  as 
the  pressure  of  the  wind  on  a  roof  has  been  shown  by  experi- 
ment to  be  normal  to  the  roof  on  the  side  from  which  it  blows, 
we  will  next  consider  the  case  of  a  triangular  truss  with  the 
load  distributed  over  one  rafter  only,  and  normal  to  the  rafter. 

There  may  be  three  cases  :  — 

i°.  When  there  is  a  roller  under  one  end,  and  the  wind 
blows  from  the  other  side. 


FIG.  80. 


148 


APPLIED   MECHANICS. 


2°.  When  there  is  a  roller  under  one  end,  and  the  wind 
tlows  from  the  side  of  the  roller. 

3°.  When  there  is  no  roller  under  either  end. 

The  last  arrangement  should  always  be  avoided  except  in 
small  and  unimportant  constructions ;  for  the  presence  of  a 
roller  under  one  end  is  necessary  to  allow  the  truss  to  change 
its  length  with  the  changes  of  temperature,  and  to  prevent  the 
stresses  that  would  occur  if  it  were  confined. 


CASE   I.  —  Using   Bow's  notation,  we  have   (Fig.  81)  the 

whole  load  represented 
in  the  diagram  by  db. 
Its  resultant  acts  at  the 
middle  of  the  rafter 
AE,  whereas  the  sup- 
porting force  at  the 
right-hand  end  is  (in 
consequence  of  the  pres- 
ence of  the  roller)  verti- 
cal. Hence,  to  find  the 
line  of  action  of  the  other 
supporting  force,  pro- 
duce the  line  of  action 
of  the  load  till  it  meets 
a  vertical  line  drawn 

through  the  roller,  and  join  their  point  of  intersection  with  the 

support  where  there  is  no  roller.     We  thus  obtain  CD  as  the 

line  of  action  of  the  left-hand  support. 

We  can   now  determine  the   magnitude  of  the  supporting 

forces  be  and  cd  by  constructing  the  triangle  bcdb  of  external 

forces. 

Now  resolve  the  normal  distributed  force  db  into  two  single 

forces  (equal  to  each  other  in  this  case),  da  and  ab  respectively, 

acting  at  the  left-hand  support  and  at  the  apex. 


FIG.  81. 


TRIANGULAR    TRUSS:     WIND   PRESSURE. 


149 


Now  proceed  to  the  left-hand  support.  We  find  four  forces 
in  equilibrium,  of  which  two  are  entirely  known  ;  viz.,  cd  and 
da:  hence,  constructing  the  quadrilateral  cdaec,  we  have  ae  as 
the  thrust  in  AE,  and  ec  as  the  tension  in  EC. 

Next  proceed  to  the  apex,  and  construct  the  triangle  of 
equilibrium  abea,  and  we  obtain  be  as  the  thrust  in  BE. 

The  triangle  bceb  is  then  the  triangle  of  equilibrium  for  the 
right-hand  sup- 
port. 

CASE  II.  - 
In  this  case 
(Fig.  82)  we  fol- 
low the  same 
method  of  pro- 
cedure, only  the 
point  of  inter- 
section of  the 
load  and  sup- 
porting forces 
is  above,  instead  of  below,  the  truss.  The  figure  explains  itself 

so  fully  that  it  is  unnecessary  to 

explain  it  here. 

CASE  III.  —  In  this  case  the 
supports  are  capable  of  exerting 
resistance  in  any  direction  what- 
ever ;  so  that,  if  any  circumstance 
should  determine  the  direction 
of  one  of  them,  that  of  the  other 
would  be  determined  also.  When  there  is  no  such  circum- 
stance, it  is  customary  to  assume  them  parallel  to  the  load 
(Fig.  83).  Making  this  assumption,  we  begin  by  dividing  the 
line  db,  which  represents  the  load,  into  two  parts,  inversely 


FIG.  8 


FIG.  83. 


150  APPLIED   MECHANICS. 

proportional  to  the  two  segments  into  which  the  line  of  action 
of  the  resultant  of  the  load  (the  dotted  line  in  the  figure) 
divides  the  line  EC.  We  thus  obtain  the  supporting  forces  be 
and  cd,  and  bcdb  is  the  triangle  of  external  forces.  We  then 
follow  the  same  method  as  in  the  preceding  cases. 

§  130.  General  Determination  of  the  Stresses  in  Roof- 
Trusses.  —  In  order  to  compute  the  stresses  in  the  different 
members  of  a  roof-truss,  it  is  necessary  first  to  know  the 
amount  and  distribution  of  the  load. 

This  consists  generally  of — » 

i°.  The  weight  of  the  truss  itself. 

2°.  The  weight  of  the  purlins,  jack-rafters,  and  superin- 
cumbent roofing,  as  the  planks,  slate,  shingles,  felt,  etc. 

3°.  The  weight  of  the  snow. 

4°.  The  weight  of  the  ceiling  of  the  room  immediately 
below  if  this  is  hung  from  the  truss,  or  the  weight  of  the 
floor  of  the  loft,  and  its  load,  if  it  be  used  as  a  room. 

5°.  The  pressure  of  the  wind ;  and  this  may  blow  from 
either  side. 

6°.  Any  accidental  load  depending  on  the  purposes  for  which 
the  building  is  used.  As  an  instance,  we  might  have  the  case 
where  a  system  of  pulleys,  by  means  of  which  heavy  weights 
are  lifted,  is  attached  to  the  roof. 

In  regard  to  the  first  two  items,  and  the  fourth,  whenever 
the  construction  is  of  importance,  the  actual  weights  should 
be  determined  and  used.  In  so  doing,  we  can  first  make  an 
approximate  computation  of  the  weight  of  the  truss,  and  use  it 
in  the  computation  of  the  stresses  ;  the  weights  of  the  ceiling 
or  of  the  floor  below  being  accurately  determined.  After  the 
stresses  in  the  different  members  have  been  ascertained  by  the 
use  of  these  loads,  and  the  necessary  dimensions  of  the  mem- 
bers determined,  we  should  compute  the  actual  weight  of  the 
truss ;  and  if  our  approximate  value  is  sufficiently  different 
from  the  true  value  to  warrant  it,  we  should  compute  again 


STA'£SS£S  IN  ROOF-TRUSSES. 


the  stresses.    This  second  computation  will,  however,  seldom  be 
necessary. 

In  making  these  computations,  the  weights  of  a  cubic  foot 
of  the  materials  used  will  be  needed ;  and  average  values  are 
given  in  the  following  table  with  sufficient  accuracy  for  the 
purpose. 


WEIGHT  OF  SOME  BUILDING  MA- 
TERIALS PER  CUBIC  FOOT. 

Pounds. 

WEIGHT  OF  SLATING  PER  SQUARE 
FOOT. 
According  to  Trautwine. 

Pounds. 

TIMBER. 

41 

\  inch  thick  on  laths      .     .    . 
\    "        "      "  i-inch  boards  . 

4-75 
6.75 

Hemlock     ••••••• 

2  C 

JL     u           «         «    jl    «           « 

7-30 

Maple     

^•j 
41 

^  "        "      "  laths      .    .    . 

7-00 

Oak,  live     

CQ 

•&  "        "      "  i-inch  boards, 

9-OO 

Oak  white  

4Q 

&  "        "      "  'i  "        "      • 

9-55 

Pine,  white  ••**••• 

'Vy 
2  C  to  "3O 

i    "        "      «  laths      .     .     . 

9-25 

Pine,  yellow,  Southern      .    . 
Spruce     ...          .    .     .    . 

45 

2  C  to  7O 

i    "        "      "  i-inch  boards, 

JL     <<           «         «    TJL   «           « 

11.25 
1  1.  80 

IRON. 

With  slating-felt  add      .     .    . 
With  |-mch  mortar  add      .     . 

ilb. 
3lbs. 

4  Co 

Iron,  wrought  ...... 

480 

NUMBER  OF  NAILS  IN  ONE  POUND 

No 

Steel  

490 

3  -penny  . 

4  CO 

OTHER  SUBSTANCES. 

80  to  90 

4      "       
6     "       

340 
I  CO 

Mortar,  hardened     .... 

IOT 

8     "       

IOO 

Snow,  freshly  fallen 

5  to   12 

10        "          .... 

60 

Snow,  compacted  by  rain  . 

I  C  to     CO 

12      "       

40 

Slate  

140  to  180 

20     "       . 

2  c 

As  to  the  weight  of  the  snow  upon  the  roof,  Stoney  recom- 
mends the  use  of  20  pounds  per  square  foot  in  moderate 
climates  ;  and  this  would  seem  to  the  writer  to  be  borne  out  by 
the  experiments  of  Trautwine  as  recorded  in  his  handbook, 


I52  APPLIED   MECHANICS. 

although  Trautwine  himself  considers  12  pounds  per  square 
foot  as  sufficient. 

§  131.  Wind  Pressure. — While  a  great  deal  of  work  has 
been  done  to  ascertain  the  direction  and  the  greatest  intensity 
of  the  pressure  of  the  wind  upon  exposed  surfaces,  as  those  of 
roofs  and  bridges,  nevertheless  the  amount  of  information  on 
the  subject  is  very  small,  inasmuch  as  but  few  experiments 
have  been  under  the  conditions  of  practice.  Before  giving  a 
summary  of  what  has  been  done  the  following  statements  will 
be  made : 

i°.  The  pressure  of  the  wind  upon  a  roof,  or  other  surface, 
is  assumed  to  be  normal  tc  the  surface  upon  which  it  blows  ; 
and  what  little  experimenting  has  been  done  upon  the  subject 
tends  to  confirm  this  view. 

2°.  Inasmuch  as  more  attempts  have  been  made  to  deter- 
mine experimentally  the  velocity  of  the  wind  than  its  pressure, 
hence  there  have  been  a  good  many  experiments  to  determine 
the  relation  between  the  velocity  and  the  pressure  upon  a  sur- 
face to  which  the  direction  of  the  wind  is  normal. 

3°.  A  few  experimenters  have  tried  to  determine  the  rela- 
tion between  the  intensity  of  the  pressure  on  a  surface  normal 
to  the  direction  of  the  wind  and  one  inclined  to  its  direction. 

4°.  While  the  above  have  been  the  investigations  most  com- 
monly pursued,  other  subjects  of  experiment  have  been— 

(a)  The  variation  of  pressure  with  density ;  (b)  with  tem- 
perature;  (c)  with  humidity;  (d)  with  the  size  of  surface 
pressed  upon  ;  (e)  with  the  shape  of  surface  pressed  upon ; 
(/)  whether  the  pressure  corresponding  to  a  certain  velocity  is 
the  same  whether  the  air  moves  against  a  body  at  rest,  or 
whether  the  body  moves  in  quiet  air. 

By  way  of  references  to  the  literature  of  the  subject  may 
be  given  the  following,  as  most  of  the  work  that  has  been 
done  is  included  in  them  or  in  other  references  which  they 
contain : 


WIND  PRESSURE.  153 


i°.  Proceedings  of  the  British  Institution  of  Civil  Engineers,  vol. 
Ixix.,  year  1882,  pages  80  to  218  inclusive. 

2°.  A.  R.  Wolff  :  Treatise  on  Windmills. 

3°.  C.  Shaler  Smith  :  Proceedings  American  Society  of  Civil  En- 
gineers, vol.  x.,  page  139. 

4°.  A.  L.  Rotch  :  Report  of  Work  of  the  Blue  Hill  Meteorological 
Observatory,  1887. 

5°.   Engineering,  Feb.  28th,  1890  :  Experiments  of  Baker. 

6°.  Engineering,  May  30,  June  6,  June  13,  1890:  Experiments  of 
O.  T.  Crosby. 

The  first  gives  an  account  of  a  very  full  discussion  of  the 
subject,  by  a  large  number  of  Engineers.  The  second  con- 
tains a  recommendation  that  the  temperature  of  the  air  be  con- 
sidered in  estimating  the  pressure.  The  fifth  gives  an  account 
of  Baker's  experiments  on  wind  pressure  in  connection  with  the 
building  of  the  Forth  Bridge. 

Before  an  account  is  given  of  the  experimental  work  that 
has  been  done,  the  following  statements  will  be  made  of  what 
are  some  of  the  methods  in  most  common  use  : 

1°.  A  great  many  engineers  very  commonly  call  from  40  to 
55  pounds  per  square  foot  the  maximum  pressure  on  a  vertical 
surface  at  right  angles  to  the  direction  of  the  wind.  One  rather 
common  practice,  in  the  case  of  bridges,  is  to  estimate  30 
pounds  per  square  foot  on  the  loaded,  or  50  pounds  per  square 
foot  on  the  unloaded  structure.  Nevertheless  pressures  of  80 
and  90  pounds  per  square  foot  have  been  registered  and  re- 
corded by  the  use  of  small  pressure-plates,  and  by  computation 
from  anemometer  records. 

2°.  By  way  of  determining  the  intensity  of  the  pressure  on 
an  inclined  surface  in  terms  of  that  on  a  surface  normal  to  the 
direction  of  the  wind,  four  methods  more  or  less  used  will  be 
enumerated  here : 

(a)  Duchemin's  formula,  which  Professor  W.  C.  Unwin 
recommends,  is  as  follows,  viz. : 


154 


APPLIED   MECHANICS. 


sin  6 


-*\  +  sin'0' 

where  /  =  intensity  of  normal  pressure  on  roof,  /,  =  intensity 
of  piessure  on  a  plane  normal  to  the  direction  of  the  wind. 
(b)  Hutton's  formula, 

p=pl  (sin  0)*.*4  &»•-*. 

Unwin  claims  that  this  and  Duchemin's  formula  give  nearly 
the  same  results  for  all  angles  of  inclination  greater  than  15°. 

The  following  table  gives  the  results  obtained  by  the  use  of 
each,  on  the  assumption  that  pl  =  40  : 


e 

Duchemin. 

Hutton. 

9 

Duchemin. 

Hutton. 

5° 

6.89 

5-10 

50° 

38.64 

38.10 

10° 

13-59 

9.60 

55° 

39-21 

39-40 

15° 

19.32 

14.20 

60° 

39-74 

40.00 

20° 

24.24 

18.40 

65° 

39-82 

40.00 

25° 

28.77 

22.6O 

70° 

39-91 

40.00 

30° 

32.00 

26.50 

75° 

39  -96 

40.00 

35° 

34-52 

30.10 

80° 

40.00 

40.00 

40° 

36.40 

33-30 

85° 

40.00 

40.00 

45° 

37-73 

36.00 

90° 

40.00 

40.00 

(c)  A  formula  very  commonly  favored,  but  which  does  not 
agree  with  any  experiments  that  have  been  made,  is 


sn 


6. 


It  gives  much  lower  results,  as  a  rule,  than  either  of  the  others, 
but  it  is  favored  by  many  because,  if  we  assume  the  wind  to 
blow  in  parallel  lines  till  it  strikes  the  surface,  and  then  to  get 
suddenly  out  of  the  way,  forming  no  eddies  on  the  back  side 
of  the  surface  and  meeting  no  lateral  resistance  on  the  front 


WIND  PRESSURE.  155 


side,  all  of  which  are  conditions  that  do  not  exist,  we  could 
then  deduce  it  as  follows: 

Assume  a  unit  surface  making  an  angle  6  with  the  direction 
of  the  wind,  the  total  pressure  on  this  surface  in  the  direction 
of  the  wind  would  be^j  sin  #;  and  by  resolving  this  into  nor- 
mal and  tangential  components  we  should  have,  for  the  former, 


(d]  Another  rule  which  is  sometimes  used,  but  which  has 
nothing  to  recommend  it,  is  to  consider  the  normal  intensity 
of  the  wind  pressure  per  square  foot  of  roof  surface  as  equal  to 
the  number  of  degrees  of  inclination  of  the  roof  to  the  hori- 
zontal. The  wind  pressure  allowed  for  by  this  rule  is  very 
excessive,  as  it  would  be  90  pounds  per  square  foot  for  a  ver- 
tical surface. 

Taking  up,  now,  the  experimental  work  that  has  been  done, 
we  will  begin  with  the  attempts  to  determine  velocities  and 
pressures,  and  the  relation  between  them. 

i°.  In  regard  to  velocities,  these  are  determined  by  using 
some  kind  of  an  anemometer,  and  in  all  these  cases  there  are 
several  difficulties  and  sources  of  error,  as  follows  : 

(a)  In  many  cases  the  anemometers  have  not  even   been 
graduated  experimentally,  but  it  has  been  assumed  outright 
that  the  velocity  of  the  air  is  just  three  times  the  linear  velocity 
of  the  cups  of  a  cup  anemometer. 

(b)  When  they  have  been  graduated,  it  has  generally  been 
done  by  attaching  them  to  the  end  of  the  arm  of  a  whirling 
machine,  which,  when  the  arm  is  long,  and  the  velocity  moder- 
ate, will  do  very  well,  but  is  the  more  inaccurate  the  shorter 
the  arm  and  the  higher  the  velocity  of  motion. 

(c)  The  wind  always  comes  in   gusts,  and   hence  the  ane- 
mometer does  not  register  the  average  velocity  of  the  wind  at 
any  one  moment,  but  that  of  the  particular  portion  that  comes 


156  APPLIED  MECHANICS. 

in  contact  with  it,  and  this  is  always  a  small  portion,  on  ac- 
count of  the  small  size  of  the  anemometer. 

(d)  In  order  to  get  an  indication  which  is  not  affected  by 
the  cross-currents  reflected  from  the  surrounding  buildings  and 
chimneys,  it  is  necessary  to  put  the  anemometer  very  high  up, 
and  then,  of  course,  we  obtain  the  indications  corresponding 
to  that  height,  which  is  greater  than  that  of  the  buildings,  and 
it  is  well  known  that  the  velocity  of  the  wind  increases  very 
considerably  with  the  height. 

Next,  as  to  the  direct  determination  of  pressure,  this  has 
usually  been  done  by  means  of  some  kind  of  pressure-plate, 
either  round  or  square,  but  of  small  size,  thus  allowing  the 
eddies  formed  on  the  back  side  of  the  plate  to  have  a  con- 
siderable effect.  The  results  obtained  by  the  use  of  different 
sizes  and  different  shapes  of  .plates  have  therefore  differed  very 
considerably ;  and  while  some  have  claimed  that  the  pressure 
per  square  foot  increases  with  the  size  of  the  surface  pressed 
upon,  it  has  been  very  thoroughly  proved  by  the  more  modern 
investigations  that  the  opposite  is  true,  and  that  the  pressure 
decreases  with  the  size. 

While  the  records  from  small  pressure-plates  have  fre- 
quently shown  very  high  pressures  per  square  foot,  as  80,  90, 
or  even  over  100  pounds  per  square  foot,  it  has  become  very 
generally  recognized  by  engineers  that  by  far  the  greater  part 
of  existing  buildings  and  bridges  would  be  overturned  by  winds 
of  such  force,  or  anywhere  near  such  force,  and  it  has  not  been 
customary  among  them  to  make  use  of  such  high  figures  for 
wind  pressure  on  bridges  and  roofs  in  computing  the  stability 
of  structures.  While  some  of  the  figures  in  general  use  have 
already  been  given,  nevertheless  the  tendency  of  modern  inves- 
tigation seems  to  be  to  obtain  rather  lower  figures.  In  this  con- 
nection it  is  well  to  refer  to  the  work  done  by  Baker  in  connec- 
tion with  the  construction  of  the  Forth  Bridge.  The  following 
description  is  taken  from  "  Engineering"  of  Feb.  28th,  1890: 


WIND   PRESSURE. 


157 


"  The  wind  pressure  to  be  provided  for  in  the  calcu- 
lations for  bridges  in  exposed  positions  is  56  Ibs.  per  square 
foot,  according  to  the  Board  of  Trade  regulations,  and  this 
twice  over  the  whole  area  of  the  girder  surface  exposed,  the 
resistance  to  such  pressure  to  be  by  dead-weight  in  the  struc- 
ture alone. 

44  The  most  violent  gales  which  have  occurred  during  the 
construction  of  the  Forth  Bridge  are  given,  with  the  pressures 
recorded  on  the  wind  gauges,  in  the  annexed  table : 


Year. 

Month 
and 
Day. 

Pressure  in  pounds  per  square  foot. 

Direction 
of 
Wind. 

Revolving 
Gauge. 

Small 
fixed 
Gauge. 

Large 
fixed 
Gauge. 

In  centre 
of  large 
Gauge. 

Right- 
hand  top 
of  large 
Gauge. 

1883 

Dec.    ii, 

33 

39 

22 

s.  w.* 

1884 

Jan.    26, 

65 

4i 

35 

s.  w.* 

1884 

Oct.    27, 

29 

23 

18 

s.  w. 

1884 

Oct.    28, 

26 

29 

!9 

s.  w. 

1885 

Mar.  20, 

30 

25 

17 

w. 

1885 

Dec.     4, 

25 

27 

19 

w. 

1886 

Mar.  31, 

26 

3i 

J9 

s.  w. 

1887 

Feb.     4, 

26 

4i 

15 

s.  w. 

1888 

Jan.      5, 

27 

16 

7 

S.  E. 

[888 

Nov.  17, 

35 

41 

27 

w. 

1889 

"          2, 

27 

34 

12 

s.  w. 

£890 

Jan.    19, 

27 

28 

16 

s.  w. 

1890 

"         21, 

26 

38 

15 

w. 

1890 

"         22, 

27 

24 

18 

231 

22 

S.  W.  by  W. 

*  These  data  are  unreliable,  owing  to  faulty  registration  by  the  indicator-needle,  as  will 
b'  presently  explained.  They  were  altered  after  this  date.  The  barometer  fell  to  27.5  inches 
ot,  <hat  occasion,  over  three  quarters  of  an  inch  within  an  hour. 


158  APPLIED  MECHANICS. 

"The  pressure-gauges,  which  were  put  up  in  the  summer 
of  1882  on  the  top  of  the  old  castle  of  Inchgarvie,  and  from 
which  daily  records  have  been  taken  throughout,  were  of  very 
simple  construction.  The  maximum  pressures  only  were  taken. 
The  most  unfavorable  direction  from  which  the  wind  pressure 
can  strike  the  bridge  is  nearly  due  east  and  west,  and  two  out 
of  the  three  gauges  were  fixed  to  face  these  directions,  while 
a  third  was  so  arranged  as  to  register  for  any  direction  of 
wind. 

"The  principal  gauge  is  a  large  board  20  feet  long  by  15 
feet  high,  or  300  square  feet  area,  set  vertically  with  its  faces  east 
and  west.  The  weight  of  this  board  is  carried  by  two  rods  sus- 
pended from  a  framework  surrounding  the  board,  and  so  ar- 
ranged as  to  offer  as  little  resistance  as  possible  to  the  passage 
of  the  wind,  in  order  not  to  create  eddies  near  the  edge  of  the 
board.  In  the  horizontal  central  axis  of  the  board  there  are 
fixed  two  pins  which  fit  into  the  lower  eyes  of  the  suspension- 
rods,  the  object  being  to  balance  the  board  as  nearly  as  pos- 
sible. Each  of  the  four  corners  of  the  board  is  held  between 
two  spiral  springs,  all  carefully  and  easily  adjusted  so  that  any 
pressure  exerted  on  either  face  will  push  it  evenly  in  the  op- 
posite direction,  but  upon  such  pressure  being  removed  the 
compressed  springs  will  force  the  board  back  to  its  normal 
position.  To  the  four  corners  four  irons  are  attached,  uniting 
in  a  pyramidal  formation  in  one  point,  whence  a  single  wire 
passes  over  a  pulley  to  the  registering  apparatus  below.  In 
order  to  ascertain  to  some  extent  how  far  great  gusts  of  wind 
are  quite  local  in  their  action,  and  exert  great  pressure  only 
upon  a  very  limited  area,  two  circular  spaces,  one  in  the  exact 
centre  and  one  in  the  right-hand  top  corner,  about  18  inches 
in  diameter,  were  cut  out  of  the  board  and  circular  plates  in- 
serted, which  could  register  independently  the  force  of  the 
wind  upon  them. 

"  By  the  side  of  the  large  square  board,  at  a  distance  of 


WIND  PRESSURE.  159 


about  8  feet,  another  gauge,  a  circular  plate  of  1.5  square  feet 
area,  facing  east  and  west,  was  fixed  up  with  separate  regis- 
tration. This  was  intended  as  a  check  upon  the  indications 
given  by  the  large  board. 

"Another  gauge  of  the  same  dimensions  as  the  last,  but 
with  the  disc  attached  to  the  short  arm  of  a  double  vane,  so 
that  it  would  face  the  wind  from  whatever  direction  it  might 
come,  was  set  up. 

"  On  one  occasion  the  small  fixed  board  appeared  to  regis- 
ter 65  pounds  to  the  square  foot — a  registration  which  caused 
no  little  alarm  and  anxiety.  Mr.  Baker  found,  upon  inves- 
tigation, that  the  registering  apparatus  was  not  in  good  order, 
and  after  adjusting  it  the  highest  pressure  recorded  was  41 
pounds. 

"  In  order  to  determine  the  effect  of  the  wind  upon  surfaces 
like  that  of  the  exposed  surface  of  the  bridge,  he  devised  an 
apparatus  which  consisted  of  a  light  wooden  rod  suspended  in 
the  middle,  so  as  to  balance  correctly,  by  a  string  from  the 
ceiling.  At  one  end  was  attached  a  cardboard  model  of  the 
surface,  the  resistance  of  which  was  to  be  tested,  and  at  the 
opposite  end  was  placed  a  sheet  of  cardboard  facing  the  same 
way  as  the  model,  so  arranged  that  by  means  of  another  and 
adjustable  sheet,  which  would  slide  in  and  out  of  the  first, 
the  surface  at  that  end  could  be  increased  or  decreased  at 
the  will  of  the  operator.  The  mode  of  working  is  for  a 
person  to  pull  it  from  its  perpendicular  position  towards 
himself,  and  then  gently  release  it,  being  careful  to  allow 
both  ends  to  go  together.  If  this  is  properly  done,  it  is  evi- 
dent that  the  rod  will  in  swinging  retain  a  position  parallel 
to  its  original  position,  supposing  that  the  model  at  one 
end  and  the  cardboard  frame  at  the  other  are  balanced  as 
to  weight,  and  that  the  two  surfaces  exposed  to  the  air 
pressure  coming  against  it  in  swinging  are  exactly  alike. 
Should  one  area  be  greater  than  the  other,  the  model  or  card- 


160  APPLIED  MECHANICS. 


board  sheet,  whichever  it  may  be,  will  b~  lagging  behind,  and 
twist  t^e  string." 

The  experiments  carried  on  in  various  ways  by  different 
people  and  at  different  times  are  generally  in  agreement  with 
each  other  and  with  the  results  of  more  elaborate  processes. 
The  information  specially  desired  was  in  regard  to  the  wind 
pressure  upon  surfaces  more  or  less  sheltered  by  those  imme- 
diately  in  front  of  them.  In  this  regard  Mr.  Baker  satisfied 
himself  that,  while  the  results  differed  very  considerably  ac- 
cording to  the  distance  apart  of  the  surfaces,  in  no  case  was 
the  area  affected  by  the  wind,  in  any  girder  which  had  two  or 
more  surfaces  exposed,  more  than  1.8  times  the  area  of  the 
surface  directly  fronting  the  wind,  and,  as  the  calculations  had 
been  made  for  twice  this  surface,  the  stresses  which  the  struc- 
ture will  receive  from  this  cause  will  be  less  than  those  pro- 
vided for. 

Next,  as  to  the  relation  between  velocity  and  pressure,  a 
great  many  formulae  have  been  devised,  to  agree  with  the 
results  of  different  experimenters.  Most  all  of  them  make  the 
pressure  proportional  to  the  square  of  the  velocity;  while 
some  add  a  term  proportional  to  the  velocity  itself,  and  when 
higher  velocities  are  reached,  as  those  usual  in  gunnery,  terms 
have  been  introduced  with  powers  of  the  velocity  higher  than 
the  second.  It  is  hardly  worth  while  to  consider  these  dif- 
ferent formulae,  as  it  is  rather  the  pressure  than  the  velocity 
that  the  engineer  is  interested  in,  and  correct  information  in 
this  regard  is  to  be  obtained  rather  from  pressure-boards  than 
from  anemometers.  Nevertheless,  it  may  be  stated  that  one 
of  the  most  usual  formulae  is  that  of  Smeaton,  and  is 


200 

where  P=  pressure  in  pounds  per  square  foot,  and  V  '=  velocity 


WIND  PRESSURE.  l6l 


in  miles  per  hour.  This  formula  agrees  very  well  with  a  num- 
ber of  experiments  that  have  been  made  where  anemometers 
have  been  used  to  determine  the  velocity,  and  small  pressure- 
plates  (say  one  square  foot)  to  determine  the  pressure  ;  thus 
this  formula  satisfies  very  well  the  experiments  made  at  the 
Blue  Hill  Meteorological  Observatory,  near  Boston,  Mass., 
U.  S.  A. 

It  was  originally  deduced  from  some  very  old  experiments 
of  Rouse  ;  and  it  agrees  with  a  good  many,  but  disagrees  with 
other  experiments.  It  is  probably  the  formula  that  has  been 
more  quoted  than  any  other. 

A  little  ought  also  to  be  said  in  regard  to  the  pressure  of 
the  wind  on  very  high  structures,  as  on  the  piers  of  high  via- 
ducts and  on  tall  chimneys.  In  this  regard  it  is  to  be  ob- 
served : 

i°.  The  pressure,  as  well  as  the  velocity  of  the  wind,  be- 
comes greater  the  higher  up  from  the  ground  the  surface  ex- 
posed is  situated. 

2°.  From  calculations  on  chimneys  that  have  stood  for  a 
long  time,  Rankine  deduced,  as  the  greatest  average  wind 
pressure  that  could  be  realized  in  the  case  of  tall  chimneys,  55 
pounds  per  square  foot. 

3°.  In  making  the  piers  of  high  viaducts,  it  would  seem 
desirable  not  to  make  them  solid,  but  to  use  only  four  up- 
rights at  the  corners  connected  by  lattice  work,  in  order  to- 
expose  a  smaller  surface  to  the  wind.  Nevertheless,  as  was  ex- 
plained, it  will  not  do  to  separate  the  structure  into  its  com- 
ponent parts,  and  to  estimate  the  pressure  on  each  part 
separately  and  then  add  the  results  together  to  get  the  total 
effect ;  but  we  really  need  some  such  experiments  as  those  of 
Baker. 

4°.  Some  old  experiments  of  Borda  bear  out  the  common 
practice  of  assuming  the  wind  pressure  on  the  surface  of  a  cir- 


1 62  APPLIED  MECHANICS 

cular  cylinder  one  half  that  which  would  exist  on  its  projection 
on  a  plane  normal  to  the  direction  of  the  wind, 

There  remains  now  only  to  refer  to  a  serial  article  by  O.  T. 
Crosby,  in  "  Engineering"  of  May  30,  June  6th,  and  June  I3th, 
1890,  containining  some  experiments  made  by  him  on  wind 
pressure  near  Baltimore,  Md.  The  first  two  numbers  contain 
rather  a  summary  of  what  has  been  done  by  others,  and  it  is 
in  the  copy  of  June  I3th  that  is  to  be  found  the  account  of  his 
own  work,  which  was  done  in  order  to  determine  the  resistances 
of  the  air  to  fast-moving  trains. 

He  used  a  whirling  arrangement  turning  about  a  vertical 
axis,  to  the  end  of  which  was  attached  a  car,  the  circumference 
through  which  the  car  moved  being  36  feet. 

In  order  to  determine  whether  the  circular  motion  produced 
any  disturbing  effect,  he  ran  a  car  having  a  cross-section  of  5.1 
square  feet  on  a  circular  track  about  two  miles  in  circumference, 
the  speed  of  the  car  being  about  50  miles  per  hour,  and  the 
results  obtained  in  this  way  agreed  very  nearly  with  those  ob- 
tained from  his  whirling  table.  The  special  peculiarity  of  his 
results  is  that  he  obtained,  by  plotting  them,  the  law  that  the 
pressure  varies  directly  as  the  first  power  of  the  velocity,  and 
not  as  the  square  or  some  higher  power;  also,  his  pressures, 
after  the  velocity  had  passed  25  or  30  miles  per  hour,  are 
much  lower  than  those  given  by  Smeaton  and  others,  the  pres- 
sure on  a  normal  plane  surface  moving  at  115  miles  per  hour 
being  about  27  pounds  per  square  foot. 

The  cars  used  were  generally  about  3  feet  long  without  the 
front.  The  fronts  attached  were:  i°.  Normal  plane  surface; 
2°.  Wedge,  base  i,  height  I  ;  3°.  Pyramid,  base  I,  height  2;  4°. 
Wedge  and  cyma,  base  I,  height  2;  5°.  Parabolic  wedge, 
base  i,  height  2. 

His  experiments  covered  a  range  of  velocities  from  30  to 
130  miles  per  hour. 


DISTRIBUTION  OF    THE   LOADS.  163 

The  law  of  the  first  powers  of  the  velocities  seems  peculiar, 
and  certainly  ought  not  to  be  accepted  without  further  cor- 
roborative evidence  ;  but  the  low  values  of  the  pressures  agree 
with  Baker's  results  and  with  the  tendency  of  the  more  modern 
investigations. 

§  132.  Approximate  Estimation  of  the  Load.  —  In  all 
important  constructions,  the  estimates  of  the  loads  should  be 
made  as  above  described.  For  smaller  constructions,  and  for 
the  purposes  of  a  preliminary  computation  in  all  cases,  we  only 
estimate  the  roof-weight  roughly ;  and  some  authors  even  as- 
sume the  wind  pressure  as  a  vertical  force. 

Trautwine  recommends  the  use  of  the  following  figures  for 
the  total  load  per  square  foot,  including  wind  and  snow,  when 
the  span  is,  75  feet  or  less  :  — 

Roof  covered  with  corrugated  iron,  unbearded     ...  28  Ibs. 

Roof  plastered  below  the  rafters 38  " 

Roof,  corrugated  iron  on  boards 31  " 

Roof  plastered  below  the  rafters 41  " 

Roof,  slate,  unboarded  or  on  laths 33  " 

Roof,  slate,  on  boards  ij  inches  thick 35  " 

Roof,  slate,  if  plastered  below  the  rafters 46  " 

Roof,  shingles  on  laths 30  " 

Roof  plastered  below  rafters  or  below  tie-beam    .     .     .  40  " 
From  75  to  100  feet,  add  4  Ibs.  to  each. 

§  133.  Distribution  of  the  Loads.  —  The  methods  for  de- 
termining the  stresses,  which  will  be  used  here,  give  correct 
results  only  when  the  loads  are  concentrated  at  joints,  and  are 
not  distributed  over  any  members  of  the  truss. 

In.  constructions  of  importance,  this  concentration  of  the 
loads  at  the  joints  should  always  be  effected  if  possible ; 
because,  when  this  is  the  case,  the  stresses  in  the  members 
of  the  truss  can  be,  if  properly  fitted,  obliged  to  resist  only 
stresses  of  direct  tension,  or  of  direct  compression  ;  but,  when 
there  is  a  load  distributed  over  any  member  of  the  truss,  that 
member,  in  addition  to  the  direct  compression  or  direct  tension, 
is  subjected  to  a  bending-stress-  The  effect  of  this  bending 


164 


APPLIED   MECHANICS. 


cannot  be  discussed  until  we  have  studied  the  theory  of  beams. 
Nevertheless,  it  is  a  fact  that  we  have  no  experimental  knowl- 
edge of  the  behavior  of  a  piece  under  combined  compression 
and  bending  ;  and  we  know  that  when  certain  pieces  are  to 
resist  bending,  in  addition  to  tension,  they  must  be  made  much 
larger  in  proportion  than  is  necessary  when  they  bear  tension 
only. 


FIG.  84. 

The  manner  in  which  this  concentration  of  the  loads  is 
effected,  is  shown  in  Fig.  84,  which  is  intended  to  represent  one 
of  a  series  of  trusses  that  supports  a  roof,  the  rafters  being  the 
two  lower  ones  in  the  figure.  Resting  on  two  consecutive 
trusses,  and  extending  from  one  to  the  other,  are  beams  called 
purlins,  which  should  be  placed  only  above  the  joints  of  the  truss, 
and  which  are  shown  in  cross-section  in  the  figure.  On  these 
purlins  are  supported  the  jack-rafters  parallel  to  the  rafters,  and 
at  sufficiently  frequent  intervals  to  support  suitably  the  plank 
and  superincumbent  roofing-materials. 

By  this  means  we  secure  that  the  entire  bending-stress  comes 
upon  the  jack-rafters  and  purlins,  and  that  the  rafters  proper 
are  subjected  only  to  a  direct  compression.  When,  however, 
the  load  is  distributed,  i.e.,  when  the  roofing  rests  directly  on  the 
rafters,  or  when  the  purlins  are  placed  at  points  other  than  the 
joints,  the  bending-stress  should  be  taken  into  account;  and 
while  the  methods  to  be  developed  here  will  give  the  stresses 


DIRECT  DETERMINATION  OF   THE  STRESSES.  165 

in  all  the  members  that  are  not  subjected  to  bending,  the  bend- 
ing-stress  may  be  largely  in  excess  of  the  direct  stress  in  those 
pieces  that  are  subjected  to  bending.  How  to  take  this  into 
account  will  be  explained  later. 

Another  important  item  to  consider  is,  that,  in  constructions 
of  importance,  a  roller  should  be  placed  under  one  end  of  the 
truss  to  allow  it  to  move  with  the  change  of  temperature  ;  as 
otherwise  some  of  the  members  will  be  either  bent,  or  at  least 
subjected  to  initial  stresses.  The  presence  of  a  roller  obliges 
the  supporting  force  at  that  point  to  be  vertical,  whether  the 
load  be  vertical  or  inclined. 

It  is  customary,  and  does  not  entail  any  appreciable  error, 
to  consider  the  weight  of  the  truss  itself,  as  well  as  that  of  the 
superincumbent  load,  as  concentrated  at  the  upper  joints ;  i.e., 
those  on  the  rafters. 

In  the  case  of  a  ceiling  on  the  room  below,  or  of  a  loft 
whose  floor  rests  on  the  lower  joints,  we  must,  of  course,  ac- 
count the  proper  load  as  resting  on  the  lower  joints. 

§134.  Direct  Determination  of  the  Stresses.  —  This,  as 
we  have  seen,  is  merely  a  question  of  equilibrium  of  forces  in 
a  plane,  where  certain  forces  acting  are  known,  and  others  are 
to  be  determined. 

As  to  the  methods  of  solution,  we  might  adopt  — 

i°.  A  graphical  solution,  laying  off  the  loads  to  scale,  and 
constructing  the  diagram  by  the  use  of  the  propositions  of 
the  polygon,  and  the  triangle  of  forces,  and  then  determining  the 
results  by  measuring  the  lines  representing  the  stresses  to 
the  same  scale. 

2°.  An  analytical  solution,  imposing  the  analytical  conditions 
of  equilibrium,  as  given  under  the  "  Composition  of  Forces," 
between  the  known  and  unknown  forces. 

3°.  A  third  method  is  to  construct  the  diagram  as  in  the 
graphical  solution,  but  then,  instead  of  determining  the  results 
by  measuring  the  resulting  lines  to  scale,  to  compute  the  un- 


1 66  APPLIED  MECHANICS. 

known  from  the  known  lines  of  the  diagram  by  the  ordinary 
methods  of  trigonometry. 

.The  first,  or  purely  graphical,  method,  is  one  which  has 
received  a  very  large  amount  of  attention  of  late  years,  and 
in  which  a  great  deal  of  progress  has  been  made.  It  is,  doubt- 
less, very  convenient  for  a  skilled  draughtsman,  and  especially 
convenient  for  one  who,  though  skilled  in  draughting,  is  not 
free  with  trigonometric  work ;  but  it  seems  to  me,  that,  when 
the  results  are  determined  by  computation  from  the  diagram, 
there  is  less  chance  of  a  slight  error  in  some  unfavorable  tri- 
angle vitiating  all  the  results.  I  am  therefore  disposed  to 
recommend  for  roof-trusses  the  third  method. 

In  the  case  of  bridge-trusses,  on  the  other  hand,  I  believe 
the  graphical  not  to  be  as  convenient  as  a  purely  analytic 
method. 

§  135.  Roof-Trusses.  —  In  what  follows,  the  graphical  solu- 
tions will  be  explained  with  very  little  reference  to  the  trigono- 
metric work,  as  that  varies  in  each  special  case,  and  to  one  who 
has  a  reasonable  familiarity  with  the  solution  of  plane  triangles, 
it  will  present  no  difficulty  ;  whereas  to  deduce  the  formulae 
for  each  case  would  complicate  matters  very  much.  Proceed- 
ing to  special  examples,  let  us  take,  first,  the  truss  shown  in 
Fig.  85,  and  let  the  vertical  load  upon  it  be  W  uniformly  dis- 
tributed over  the  top  of  the  roof,  the  purlins  being  at  the  joints 
on  the  rafters. 

The  loads  at  the  several  joints  will  then  be  as  follows,  viz. 
(Fig.  85*),  — 

ab  =  kl  =  ~,    be  =  cd  =  de  =  ef  =  fg  =  gh  =  hk  =  ~. 
16  8 

Then  the  supporting  forces  will  be 

lm  =  ma  =  — . 

2 

We  thus  have,  as  polygon  of  external  forces,  abcdefghklma. 


ROOF-TRUSSES. 


I67 


Now  proceed  to  either  support,  say,  the  left-hand  one ;  and 
there  we  have  the  two  forces  ab  and  ma  known,  while  by  and 
ym  are  unknown.  We  then  construct 
the  quadrilateral  maby  in  the  figure,  and 
thus  determine  by  and  ym.  As  to  whether 


FIG.  850. 


FIG.  8s<5. 


dbcde 


FIG.  85. 

these  represent  thrust  or  tension, 
we  need  only  remember  that  they 
are  the  forces  exerted  by  the  re- 
spective bars  at  the  joints :  and,  since  by  is  directed  away  from 
the  bar  BY,  this  bar  is  in  compression;   whereas,  ym  being 
directed  towards  the  bar  YM,  that  bar  is  in  tension. 


l68  APPLIED   MECHANICS. 

Having  determined  these  two  stresses,  we  next  proceed  to 
another  joint,  where  we  have  only  two  unknown  forces.  Take 
the  joint  at  which  the  load  be  acts,  and  we  have  as  known 
quantities  the  load  be,  and  also  the  force  exerted  by  the  bar 
YB,  which  is  in  compression.  This  force  is  now  directed  away 
from  the  bar,  and  hence  is  represented  by  yb.  The  unknown 
forces  are  the  stresses  in  CX  and  XY.  Hence  we  construct 
the  quadrilateral  cxybc  ;  and  we  thus  determine  the  stresses  in 
CX  and  XY  as  ex  and  xyt  both  being  thrusts. 

Next  proceed  to  the  joint  YXW,  and  construct  the  quadri- 
lateral myxwm,  and  thus  determine  the  tension  xw  and  the 
tension  wm. 

Next  proceed  to  the  joint  where  cd  acts,  and  so  on.  We 
thus  obtain  the  diagram  (Fig.  85*2)  giving  all  the  stresses. 

The  truss  in  the  figure  was  constructed  with  an  angle  of  30* 
at  the  base,  and  hence  gives  special  values  in  accordance  with 
that  angle. 

In  order  to  show  how  we  may  compute  the  stresses  from  the 
diagram,  the  computation  will  be  given. 

From  triangle  bmy,  we  have  bm  =  -£•  W 

10 


ym  =     -Wcot  30°  = 

16  16 


by  =  --^cosec  30°  =     w  =  ky. 
1  6  8 


From  the  triangle  umc,  we  have  cm  —  —  W, 

16 


um  =          w 
16 


ROOF-TRUSSES. 


169 


yx  —  yw  sec  30 


- 
16 


=  (  ^ 

\i6 


=  —  =  xv  =  vt, 


i6 


256  256 


, 
8 


ex  =  wm  sec 


256  256 


vd  =  urn  sec  30°  =  W\  -4 

\  16      /  y/         8 


4  ' 


Hence  we  shall  have  for  the  stresses,  — 


RAFTERS 

(compression)  . 

VERTICALS 

(tension). 

by     =  kn 
ex     =  ho 

=       \W. 

xw  =  op 

W 
16* 

dv     =  gq 
ct      =fs 

=  \Z. 

vu   =  qr 

:  T' 

HORIZONTAL  TIES  (tension). 

_4/7 

ts     =                                    1  07 
8 

DIAGONAL  BRACES  (compression). 

my    =  mn 

16 

xy    =  0« 

-  if 

8  ' 

mw  =  mp 

=      g       .- 

o/z;  =  qp 

~"i6 

mu  =  mr 

16 

fu    =  sr 

**  8 

1/0  APPLIED   MECHANICS. 

Next,  as  to  the  stresses  due  to  wind  pressure,  we  will  sup- 
pose that  there  is  a  roller  under  the  left-hand  end  of  the  truss, 
and  none  under  the  right-hand  end ;  and  we  will  proceed  to 
determine  the  stresses  due  to  wind  pressure. 

First,  suppose  the  wind  to  blow  from  the  left-hand  side  of 
rhe  truss,  and  let  the  total  wind  pressure  be  (Fig.  8$b)  af=  W^. 
The  resultant,  of  course,  acts  along  the  dotted  line  drawn  per- 
pendicular to  the  left-hand  rafter  at  its  middle  point,  as  shown 
in  Fig.  85. 

The  left-hand  supporting  force  will  be  vertical :  hence,  pro- 
ducing the  above-described  dotted  line,  and  a  vertical  through 
the  roller  to  their  intersection,  and  joining  this  point  with  the 
right-hand  end  of  the  truss,  we  have  the  direction  of  the  right- 
hand  supporting  force.  In  this  case,  since  the  angle  of  the 
truss  is  30°,  the  line  of  action  of  the  right-hand  supporting 
force  coincides  in  direction  with  the  right-hand  rafter.  We 
now  construct  the  triangle  of  external  forces  afmy  and  we 
obtain  the  supporting  forces  fm  and  ma.  We  then  have,  as 
the  loads  at  the  joints,' 

ab  —  — -  =  ef, 


be  =  -      =  cd  =  de. 

4 

Then  proceed  as  before  to  the  left-hand  joint ;  and  we  find  that 
two  of  the  four  forces  acting  there  are  known,  viz.,  ma  and  ab, 
and  two  are  unknown,  viz.,  the  stresses  in  .Z?  Fand  YM.  Then 
construct  the  quadrilateral  mabym,  and  we  have  the  stresses  by 
and  ym  ;  the  first  being  compression  and  the  second  tension, 
as  shown  by  reasoning  similar  to  that  previously  adopted. 

Then  pass  to  the  next  joint  on  the  rafter,  and  construct  the 
quadrilateral  ybcxy,  where  yb  and  be  are  already  known,  and  we 
obtain  ex  and  xy  ;  and  so  proceed  as  before  from  joint  to  joint, 


ROOF-TRUSSES    WITH  LOADS  AT  LOWER  JOINTS.       I/T 

remembering,  that,  in  order  to  be  able  to  construct  the  polygon 
of  forces  in  each  case,  it  is  necessary  that  only  two  of  the  forces 
acting  should  be  unknown. 

When  the  wind  blows  from  the  other  side,  we  shall  obtain 
the  diagram  shown  in  Fig.  85^. 

After  having  determined  the  stresses  from  the  vertical  load 
diagram  and  those  from  the  two  wind  diagrams,  we  should,  in 
order  to  obtain  the  greatest  stress  that  can  come  on  any  one 
member  of  the  truss,  add  to  the  stress  due  to  the  vertical  load 
the  greater  of  the  stresses  due  to  the  wind  pressure. 

§  136.  Roof-Truss   with    Loads   at   Lower  Joints.  —  In 
Fig.  86  is  drawn  a  stress  diagram 
for  the  truss  shown  in  Fig.  84  on 
the  supposition  that  there  is  also    X. 
a  load  on  the  lower  joints.     In 
this  case  let  W  be  the  whole  load 
of  the  truss,  except  the  ceiling, 
^  the  weight  of  the  ceiling 


and 

below  ;  the  latter  being  supported 

a,t  the  lower  joints    and  on   the 

two  extreme  vertical   suspension  FlG-  86- 

rods.     Then  will  the  loads  at  the  joints  be  as  follows;  viz.,— 


ab    = 

be    =   \( 
cd    =   \W 
mn 


=  rq 


=  kl, 

=  gh  =  de  =  jfe 

=  on  =  qp  =  op. 


Observe  that  there  is  no  joint  at  the  lower  end  of  either  of  the 
end  suspension  rods,  but  that  whatever  load  is  supported  by 
these  is  hung  directly  from  the  upper  joints,  where  be  and  hk  act 
We  have  also  for  each  of  the  supporting  forces  Im  and  ra 


1/2  APPLIED   MECHANICS. 

Hence  we  have,  for  the  polygon  of  external  forces, 
abcdefghklm  nopqra, 

which  is  all  in  one  straight  line,  and  which  laps  over  on 
itself. 

In  constructing  the  diagram,  we  then  proceed  in  the  same 
way  as  heretofore. 

§  137.  General  Remarks.  —  As  to  the  course  to  be  pursued 
in  general,  we  may  lay  down  the  following  directions  :  — 

I  °.  Determine  all  the  external  forces  ;  in  other  words,  the  loads 
being  known,  determine  the  supporting  forces. 

2°.  Construct  the  polygon  of  forces  for  each  joint  of  the  truss, 
beginning  at  some  joint  where  only  two  of  the  forces  acting  at 
that  joint  are  unknown.  This  is  usually  the  case  at  the  support. 
Then  proceed  from  joint  to  joint,  bearing  in  mind  that  we  can 
only  determine  the  polygon  of  forces  when  the  magnitudes  of 
all  but  two  sides  are  known. 

3°.  Adopt  a  certain  direction  of  rotation,  and  adhere  to  it 
throughout;  i.e.,  if  we  proceed  in  right-handed  rotation  at  one 
joint,  we  must  do  the  same  at  all,  and  we  shall  thus  obtain  neat 
and  convenient  figures. 

4°.  Observe  that  the  stresses  obtained  are  the  forces  exerted 
by  the  bars  under  consideration,  and  that  these  are  thrusts  when 
they  act  away  from  the  bars,  and  tensions  when  they  are  directed 
towards  the  bars. 

We  will  next  take  some  examples  of  roof-trusses,  and  con- 
struct the  diagrams  of  some  of  them,  calling  attention  only  to 
special  peculiarities  in  those  cases  where  they  exist. 

It  will  be  assumed  that  the  student  can  make  the  trigono- 
metric computations  from  the  diagram. 

The  scale  of  load  and  wind  diagram  will  not  always  be  the 
same ;  and  the  stress  diagrams  will  in  general  be  smaller  than 
is  advisable  in  using  them,  and  very  much  too  small  if  the 


ROOF-TRUSSES    WITH  LOADS  AT  LOWER  JOINTS.        173 

results  were  to  be  obtained  by  a  purely  graphical  process  with- 
out any  computation. 

The  loads  will  in  all  cases  be  assumed  to  be  distributed 
uniformly  over  the  jack-rafters,  or,  in  other  words,  concen- 
trated at  the  joints. 

Those  cases  where  no  stress  diagram  is  drawn  may  be  con- 
sidered as  problems  to  be  solved. 


FIG.  87. 


FIG.  870. 


FIG.  87*. 


174 


APPLIED   MECHANICS. 


PIG.  88. 


FIG.  883. 


abcdl 


FIG. 


R O OF- TR USSES    WITH  LOADS  A  %  L O  WER  JOINTS.       I  7 5 


FIG.  89. 


FIG.  8ga. 


FIG.  90. 


FIG. 


a 
b 

c    A 


X 


FIG.  92. 


FIG.  92*. 


FIG.  93. 


FIG.  93a. 


APPLIED   MECHANICS. 


§138.  Hammer- Beam  Truss  (Fig.  94). — This  form  of 
truss  is  frequently  used  in  constructions  where  architectural 
effect  is  the  principal  consideration  rather  than  strength.  It 
is  not  an  advantageous  form  from  the  point  of  view  of  strength, 


FIG. 


FIG.  94. 


FIG.  944. 


FIG.  94,:. 


for  the  absence  of  a  tie-rod  joining  the  two  lower  joints  causes 
a  tendency  to  spread  out  at  the  base,  which  tendency  is  usually 
counteracted  by  *the  horizontal  thrust  furnished  by  the  but 
tresses  against  which  it  is  supported. 


HAMMER-BEAM   TRUSS.  177 

When  such  a  thrust  is  furnished  (or  were  there  a  tie-rod), 
and  the  load  is  symmetrical  and  vertical,  the  bars  are  not  all 
needed,  and  some  of  them  are  left  without  any  stress.  In 
the  case  in  hand,  it  will  be  found  that  UV,  VM,  MQ,  and  QR 
are  not  needed.  We  must  also  observe  that  the  effect  of  the 
curved  members  MY,  MV,  MQ,  and  MAT  on  the  other  parts  of 
the  truss  is  just  the  same  as  though  they  were  straight,  as 
shown  in  the  dotted  lines.  The  curved  piece,  of  course,  has  to 
be  subjected  to  a  bending-stress  in  order  to  resist  the  stress 
acting  upon  it.  If,  as  is  generally  the  case,  the  abutments  are 
capable  of  furnishing  all  the  horizontal  thrust  needed,  it  will 
first  be  necessary  to  ascertain  how  much  they  will  be  called 
upon  to  furnish.  To  do  this,  observe  that  we  have  really  a  truss 
similar  to  that  shown  in  Fig.  92,  supported  on  two  inclined 
framed  struts,  of  which  the  lines  of  resistance  are  the  dotted 
lines  (Fig.  94)  I  4  and  7  8,  and  that,  under  a  symmetrical  load, 
this  polygonal  frame  will  be  in  equilibrium,  and,  moreover,  the 
curved  pieces  MV  and  MQ  will  be  without  stress,  these  only- 
being  of  use  to  resist  unsymmetrical  loads,  as  the  snow  or 
wind. 

Let  the  whole  load,  concentrated  by  means  of  the  purlins 
at  the  joints  of  the  rafters,  be  W.  Then  will  the  truss  467  have 

W 

to  bear  \  W,  and  this  will  give  —  to  be  supported  at  each  of 

4 
the  points  4  and  7.     Moreover,  on  the  space  2  4  is  distributed 

— ,  which  has,  as  far  as  overturning  the  strut  is  concerned,  the 
4 

W  W 

same  effect  as  —  at  2,  and  —  at  4.     Hence  the  load  to  be  sup- 
8  8 

ported  at  4  by  the  inclined  strut  is  a  vertical   load  equal   to 

(i  +  J) w  —  1 w-   We  may then  find  the  force  that  must  be 

furnished  by  the  abutment,  or  by  the  tie-rod,  in  either  of  the 
two  following  ways  :  — 


178  APPLIED   MECHANICS 

i°.  By  constructing  the  triangle  ySe  (Fig.  94*2),  with  78  = 
|  W,  ye  ||  14,  and  eS  parallel  to  the  horizontal  thrust  of  the  abut- 
ment ;  then  will  y&  be  the  triangle  of  forces  at  I,  and  eS  will  be 
the  thrust  at  i. 

2°.  Multiply  f  W  by  the  perpendicular  distance  from  4  to 
i  2,  and  divide  by  the  height  of  4  above  I  8  for  the  thrust  of  the 
abutment ;  in  other  words,  take  moments  about  the  point  i. 

Now,  to  construct  the  diagram  of  stresses,  let,  in  Fig.  94^, 
the  loads  be 

ab,  be,  cd,  </<?,  ef,fg,  gh,  hk,  and  klt 
and  let 

lz  =  za  =  \W 

be  the  vertical  component  of  the  supporting  force ;  let  zm  be 
the  thrust  of  the  abutment :  then  will  Im  and  ma  be  the  real 
supporting  forces ;  and  we  shall  have,  for  polygon  of  external 
forces, 

abcdefghklma. 

Then,  proceeding  to  the  joint  i,  we  obtain,  for  polygon  of  forces, 

maym  ; 

and,  proceeding  from  joint  to  joint,  we  obtain  the  stresses  in  all 
the  members  of  the  truss,  as  shown  in  Fig.  94^. 

It  will  be  noticed  that  UV  and  RQ  are  also  free  from 
stress. 

If  we  had  no  horizontal  thrust  from  the  abutment,  and  the 
supporting  forces  were  vertical,  the  members  MV  and  MQ 
would  be  called  into  action,  and  J/Fand  MN  would  be  inactive. 
To  exhibit  this  case,  I  have  drawn  diagram  94/7,  which  shows 
the  stresses  that  would  then  be  developed.  A  Fand  NL  would 
become  merely  part  of  the  supports. 

In  this  latter  case  the  stresses  are  generally  much  greater 
than  in  the  former,  and  a  stress  is  developed  in  UV. 


SCISSOR-BEAM   TRUSS. 


§  139.  Hammer-Beam  Truss:  Wind  Pressure.  —  Fig.  95 
shows  the  stress  diagram  of  the  hammer-beam  truss  for  wind 
pressure  when  there  is  no  roller  under  either  end,  and  when 
the  wind  blows  from  the  left.  A  similar  diagram  would  give  the 
stresses  when  it  blows  from  the  right. 


FIG.  95. 


FIG. 


The  cases  when  there  is  a  roller  are  not  drawn  :  the  student 
may  construct  them  for  himself. 

§140.  Scissor-Beam  Truss.  —  We  have  already  discussed 
two  forms  of  scissor-beam  truss 
in    Figs.  90    and    91.      These 
trusses  having  the  right  number 
of  parts,  their  diagrams  present 
no  difficulty.     Another  form  of     A 
the  scissor-beam  truss  is  shown 
in  Fig.  96,  and  its  diagram  pre- 
sents no  difficulty. 

The  only  peculiarity  to  be  noticed  is,  that,  after  having  coa 
structed  the  polygon  of  external  forces, 

abcdefma, 

we  cannot  proceed  to  construct  the  polygon  of  equilibrium  for 
one  of  the  supports,  because  there  are  three  unknown  forces 


FlG- 


FlG-  ^ 


1 8o 


APPLIED   MECHANICS. 


there.  We  therefore  begin  at  the  apex  CD,  and  construct  the 
triangle  of  forces  cdl  for  this  point ;  then  proceed  to  joint  CB, 
and  construct  the  quadrilateral 

bclkb; 
then  proceed  to  the  left-hand  support,  and  obtain 

mabkgm  ; 
and  so  continue. 

§  141.     Scissor-Beam  Truss  -without   Horizontal  Tie.  — 

Very  often  the  scissor-beam  truss  is  constructed  without  any 
horizontal  tie,  in  which  case  it  has  the  appearance  of  Fig.  97, 
where  there  is  sometimes  a  pin  at  GKLH  and  sometimes  not. 


FIG.  gja. 


FIG.  97. 


FIG.  97<5. 


FIG.  97c. 


In  this  case,  if  the  abutments  are  capable  of  furnishing  hori- 
zontal thrust  to  take  the  place  of  the  horizontal  tie  of  Fig.  96, 
we  are  reduced  back  to  that  case.  If  the  abutments  are  not 
capable  of  furnishing  horizontal  thrust,  we  are  then  relying  on 
the  stiffness  of  the  rafters  to  prevent  the  deformation  of  the 
truss  ;  for,  were  the  points  BC  and  DE  really  joints,  with  pins, 
the  deformation  would  take  place,  as  shown  in  Fig.  97^  or  Fig. 
97^,  according  as  the  two  inclined  ties  were  each  made  in  one 
piece  or  in  two  (i.e.,  according  as  they  are  not  pinned  together 
at  KH,  or  as  they  are  pinned).  This  necessity  of  depending 
on  the  stiffness  of  the  rafters,  and  the  liability  to  deformation 
if  they  had  joints  at  their  middle  points,  become  apparent  as 
soon  as  we  attempt  to  draw  the  diagram.  Such  an  attempt  is 


SCISSOR-BEAM   TRUSS    WITHOUT  HORIZONTAL    TIE.      l8l 

made  in  Fig.  97^,  where  abcdefga  is  the  polygon  of  external 
forces,  gabkg  the  polygon  of  stresses  for  the  left-hand  support, 
kbclk  that  for  joint  BC.  Then,  on  proceeding  to  draw  the  tri- 
angle of  stresses  for  the  vertex,  we  find  that  the  line  joining  d 
and  /  is  not  parallel  to  DL,  and  hence  that  the  truss  is  not 
stable.  We  ought,  however,  in  this  latter  case,  when  the  sup- 
porting forces  are  vertical,  and  when  we  rely  upon  the  stiffness 
of  the  rafters  to  prevent  deformation,  to  be  able  to  determine 
the  direct  stresses  in  the  bars ;  and  for  this  we  will  employ  an 
analytical  instead  of  a  graphical  method,  as  being  the  most  con- 
venient in  this  case. 

Let  us  assume  that  there  is  no  pin  at  the  intersection  of  the 
two  ties,  and  that  the  two  rafters  are  inclined  at  an  angle  of  45° 
to  the  horizon. 

We  then  have,  if  W  =  the  entire  load,  and  a  =  angle 
between  BK  and  KG, 

w  w 

ab  =  cf  =  — ,      be  ^  cd  =  de  =  — , 
8  4 

T  2 

tana  =  4,     sin  a  =  — ,     cos  a  =  — , 

Vs  ^s 

Let  x  be  the  stress  in  each  tie,  and  let  y  =  cl  —  dl  =  thrust 
in  each  upper  half  of  the  rafters. 

Then  we  must  observe  that  the  rafter  has,  in  addition  to  its 
direct  stresses,  a  tendency  to  bend,  due  to  a  normal  load  at  the 
middle,  this  normal  load  being  equal  to  the  sum  of  the  normal 
components  of  be  and  of  x>  when  these  are  resolved  along  and 
normal  to  the  rafter.  Hence 


normal  load  =  x  cos  a  -\ sin  45 

4 


This,  resolved  into  components  acting  at  each  end  of  the  rafterr 
gives  a  normal  downward  force  at  each  end  equal  to 

-f- 


I  82  APPLIED   MECHANICS. 

Hence,  resolving  all  the  forces  acting  at  the  left-hand  support 
into  components  along  and  at  right  angles  to  the  rafter,  and 
imposing  the  condition  of  equilibrium  that  the  algebraic  sum 
of  their  normal  components  shall  equal  zero,  we  have,  if  we  call 
upward  forces  positive, 

-f  JFsin45°  —  (%xcosa  +  ^-fFsin45°)  —  #sina  =  o;     (i) 
but,  since 


we  have  from  (i) 

W 
2#sina  =  —  sin  45° 

4 

W  •        o 

/.    jfsma    =  —  sin  45 
8 


(„ 

Then,  proceeding  to  the  apex  of  the  roof,  we  have  that  the  load 

,      W 
cd  =  — 

4 
gives,  when  resolved  along  the  two  rafters,  a  stress  in  each 

equal  to 


4 

Hence  the  load  to  be  supported  in  a  direction  normal  to  the 
rafter  at  the  apex  is 

—  sin  45°  -f-  (^  cos  a  -\  --  sin  45°). 
4  8 

Hence,  substituting  for  x  its  value,  we  have 

y  =  cl=dl=  5Tsin45°.  (3) 

Then,  proceeding  to  the  left-hand  support,  and  equating  to  zero 
the  algebraic  sum  of  the  components  along  the  rafter,  we  have 

bk  =  (ga  —  0£)cos45°  ~f~  -^cosa 

°  -f  JWsii^0  =  f  ^5^45°.        (4) 


SCISSOR-BEAM   TRUSS    WITHOUT  HORIZONTAL    TIE.      183 


We  have  thus  determined  in  (2),  (3),  and  (4)  the  values  of  xy  y, 
and  bk  —  eh. 

By  way  of  verification,  proceed  to  the  middle  of  the  left- 
hand  rafter,  and  we  find  the  algebraic  sum  of  the  components 
of  be  and  x  along  the  rafter  to  be 


and  this  is  the  difference  between  bk  and  cl,  as  it  should  be. 

We  have  thus  obtained  the  direct  stresses  ;  and  we  have,  in 
addition,  that  the  rafter  itself  is  also  subjected  to  a  bending- 
moment  from  a  normal  load  at  the  centre,  this  load  being  equal 
to 

xcosa  H  --  sin  45°  =   —sin  45°. 
4  2 

How  to  take  this  into  account  will  be  explained  under  the 
"  Theory  of  Beams." 

§142.  Examples.  —  The  following  figures  of  roof  -trusses 
may  be  considered  as  a  set  of  examples,  for  which  the  stress 
diagrams  are  to  be  worked  out. 

Observe,  that,  wherever  there  is  a  joint,  the  truss  is  to  be 
supposed  perfectly  flexible,  i.e.,  free  to  turn  around  a  pin. 


FIG.  98. 


FIG.  99- 


FIG.  too. 


FIG.  101. 


FIG.  102. 


FIG.  103. 


FIG.  104. 


FIG. 


FIG.  106. 


FIG.  107. 


FIG.  108 


1 84  APPLIED  MECHANICS. 


CHAPTER   IV. 
BRIDGE-TRUSSES. 

§  143.  Method  of  Sections.  —  It  is  perfectly  possible  to 
determine  the  stresses  in  the  members  of  a  bridge-truss 
graphically,  or  by  any  methods  that  are  used  for  roof-trusses. 

In  this  work  an  analytical  method  will  be  used  ;  i.e.,  a  method 
of  sections.  This  method  involves  the  use  of  the  analytical  con- 
ditions of  equilibrium  for  forces  in  a  plane  explained  in  §  63. 
These  are  as  follows  ;  viz.,  — 

If  a  set  of  forces  in  a  plane,  which  are  in  equilibrium,  be 
resolved  into  components  in  two  directions  at  right  angles  to 
each  other,  then  — 

i°.  The  algebraic  sum  of  the  components  in  one  of  these 
directions  must  be  zero. 

2°.  The  algebraic  sum  of  the  components  in  the  other  of 
these  directions  must  be  zero. 

3°.  The  algebraic  sum  of  the  moments  of  the  forces  about 
any  axis  perpendicular  to  the  plane  of  the  forces  must  be  zero. 

Assume,  now,  a  bridge-truss  (Figs.  109,  no,  in,  112,  pages 
186  and  187)  loaded  at  a  part  or  all  of  the  joints.  Conceive  a 
vertical  section  ab  cutting  the  horizontal  members  6-8  and  7~9 
and  the  diagonal  7-8,  and  dividing  the  truss  into  two  parts. 
Then  the  forces  acting  on  either  part  must  be  in  equilibrium, 
in  other  words,  the  external  forces,  loads,  and  supporting  forces, 
acting  on  one  part,  must  be  balanced  by  the  stresses  in  the 
members  cut  by  the  section  ;  i.e.,  by  the  forces  exerted  by  the 
other  part  of  the  truss  on  the  part  under  consideration.  Hence 
we  must  have  the  three  following  conditions ;  viz.,  — - 


SHEARING-FORCE   AND   BENDING-MOMENT.  .185 

i°.  The  algebraic  sum  of  the  vertical  components  of  the 
above-mentioned  forces  must  be  zero, 

2°.  The  algebraic  sum  of  the  horizontal  components  of  these 
forces  must  be  zero. 

3°.  The  algebraic  sum  of  the  moments  of  these  forces  about 
any  axis  perpendicular  to  the  plane  of  the  truss  must  be  zero. 

§  144.  Shearing-Force  and  Bending-Moment.  —  Assum- 
ing all  the  loads  and  supporting  forces  to  be  vertical,  we  shall 
have  the  following  as  definitions. 

The  Shearing-Force  at  any  section  is  the  force  with  which 
the  part  of  the  girder  on  one  side  of  the  section  tends  to  slide 
by  the  part  on  the  other  side. 

In  a  girder  free  at  one  end,  it  is  equal  to  the  sum  of  the 
loads  between  the  section  and  the  free  end. 

In  a  girder  supported  at  both  ends,  it  is  equal  in  magnitude 
to  the  difference  between  the  supporting  force  at  either  end, 
and  the  sum  of  the  loads  between  the  section  and  that  support- 
ing force. 

The  Bending-Moment  at  any  section  is  the  resultant  moment 
of  the  external  forces  acting  on  the  part  of  the  girder  to  one  side 
of  the  section,  tending  to  rotate  that  part  of  the  girder  around 
a  horizontal  axis  lying  in  the  plane  of  the  section. 

In  a  girder  free  at  one  end,  it  is  equal  to  the  sum  of  the 
moments  of  the  loads  between  the  section  and  the  free  end, 
about  a  horizontal  axis  in  the  section. 

In  a  girder  supported  at  both  ends,  it  is  the  difference  be- 
tween the  moment  of  either  supporting  force,  and  the  sum  of 
the  moments  of  the  loads  between  the  section  and  that  sup- 
port ;  all  the  moments  being  taken  about  a  horizontal  axis  in 
the  section. 

§  145.  Use  of  Shearing-Force  and  Bending-Moment.  — 
The  three  conditions  stated  in  §  143  may  be  expressed  as  fol- 
lows :  — 

i°.  The  algebraic  sum  of  the  horizontal  components  of  the 
stresses  in  the  members  cut  by  the  section  must  be  zero. 


1 86 


APPLIED  MECHANICS. 


2°.  The  algebraic  sum  of  the  vertical  components  of  the 
stresses  in  the  members  cut  by  the  section  must  balance  the 
shearing-force. 

3°.  The  algebraic  sum  of  the  moments  of  the  stresses  in 
the  members  cut  by  the  section,  about  any  axis  perpendicular  to 
the  plane  of  the  truss,  and  lying  in  the  plane  of  the  section, 
must  balance  the  bending-moment  at  the  section. 

As  the  conditions  of  equilibrium  are  three  in  number,  they 
will  enable  us  to  determine  the  stresses  in  the  members,  pro- 
vided the  section  does  not  cut  more  than  three ;  and  this 
determination  will  require  the  solution  of  three  simultaneous 
equations  of  the  first  degree  with  three  unknown  quantities 
(the  stresses  in  the  three  members). 

By  a  little  care,  however,  in  choosing  the  section,  we  can 
very  much  simplify  the  operations,  and  reduce  our  work  to  the 
solution  of  one  equation  with  only  one  unknown  quantity ;  the 
proper  choice  of  the  section  taking  the  place  of  the  elimination. 

§146.  Examples  of  Bridge-Trusses.  —  Figs.  109-1 12  rep- 
resent two  common  kinds  of  bridge-trusses  :  in  the  first  two 

the  braces  are  all 

i    3     5  _7]ajk.n..i3   45   17  19  21  23    25  27   29       diagonal,    in     the 

last  two  they  are 
partly  vertical  and 
partly  diagonal. 

The  first  two  are  called  Warren  girders,  or  half-lattice  girders ; 
since  there  is  only  one  system  of  bracing, 
as  in  the  figures.  When,  on  the  other 
hand,  there  are  more  than  one  system,  so 
that  the  diagonals  cross  each  other,  they 
are  called  lattice  girders. 

§  147.  General  Outline  of  the  Steps 
to  be  taken  in  determining  the  Stresses 
in  a  Bridge-Truss  under  a  Fixed  Load. 

i°.  If  the  truss  is  supported  at  both  ends,  find  the  sup- 
porting forces. 


VVV\/K/V\/\/\A/\/\/\A/ 

2 4      6   b\  8  *|  10     12     14     16     18    20     22     24      2628 
FIG.  109. 


1357 

a  91  a-  11      13 

vw 

2157 

2466 

8  £j  10     13 

FIG.  no. 


DETERMINING    THE  STRESSES  IN  A    BRIDGE-TRUSS.   1 87 


2°.  Assume,  in  all  cases,  a  section,  in  such  a  manner  as  not 
to  cut  more  than  three  members  if  possible,  or,  rather,  three 
of  those  that 

1      13      15      17      19     21     23      25      27      28 

XlXIXbd/l/l/1// 


brought 


7  a 


\ 


10      12      14      16      18     20      22      24      26 


FlG- 


2      4       6 

R 

10     12     14 

/\/]/\, 

/ 

/MX 

1357 

9 

11    13 

FIG. 


are 

into  action 
by  the  loads 
on  the  truss  ; 
and  it  will 

save  labor  if  we  assume  the  section  so  as  to  cut  two  of  the 

three  very  near  their  point  of  inter- 
section. 

3°.  Find  the  shearing-force  at  the 
section. 

4°.  Find   the   bending-moment   at 
the  section. 

5°.  Impose  the  analytical  conditions  of  equilibrium  on  all 
the  forces  acting  on  the  part  of  the  girder  to  one  side  of  the 
section,  —  the  part  between  the  section  and  the  free  end  when 
the  girder  is  free  at  one  end,  or  either  part  when  it  is  supported 
at  both  ends. 

In  the  cases  shown  in  Figs.  109  and  no,  we  may  describe 
the  process  as  follows  ;  viz.,  — 

(a)  Find  the  stress  in  the  diagonal  from  the  fact,  that  (since 
the  stress  in  the  diagonal   is  the  only  one  that  has  a  vertical 
component  at  the  section)  the  vertical  component  of  the  stress 
in  the  diagonal  must  balance  the  shearing-force. 

(b)  Take  moments  about  the  point  of  intersection  of  the 
diagonal  and  horizontal  chord  near  which  the  section  is  taken  ; 
then  the  stresses  in  those  members  will  have  no  moment,  so 
that   the  moment    of   the   stress  in   the  other  horizontal  must 
balance  the  bending-moment  at  the  section.     Hence  the  stress 
in  the  horizontal  will  be  found  by  dividing  the  bending-moment 
at  the  section  by  the  height  of  the  girder. 

The  above  will  be  best  illustrated  by  some  examples. 


I  88  APPLIED   MECHANICS. 

EXAMPLE  I.  —  Given  the  semi-girder  shown  in  Fig.  no, 
loaded  at  joint  13  with  4000  pounds,  and  at  each  of  the  joints 
l>  3>  5>  7>  9»  and  ii  with  8000  pounds.  Suppose  the  length  of 
each  chord  and  each  diagonal  to  be  5  feet.  Required  the  stress 
in  each  member. 

Solution.  —  For  the  purpose  of  explaining  the  method  of 
procedure,  we  will  suppose  that  we  desire  to  find  first  the 
stresses  in  8-10  and  9-10. 

Assume  a  vertical  section  very  near  the  joint  9,  but  to  the 
right  of  it,  so  that  it  shall  cut  both  8-10  and  9-10. 

If,  now,  the  truss  were  actually  separated  into  two  parts  at 
this  section,  the  right-hand  part  would,  in  consequence  of  the 
loads  acting  on  it,  separate  from  the  other  part.  This  tendency 
to  separate  is  counteracted  by  the  following  three  forces  :  — 

i°.  The  pull  exerted  by  the  part  <$-x  of  the  bar  9-11  on  the 
part  x-\\  of  the  same  bar. 

2°.  The  thrust  exerted  by  the  part  8-2  of  the  bar  8-10  on 
the  part  ^-10  of  the  same  bar. 

3°.  The  pull  exerted  by  the  part  9-7  of  the  bar  9-10  on  the 
part  y-io  of  the  same  bar. 

The  shearing-force  at  this  section  is 

8000  -f-  4000  =  12000  Ibs., 

and  this  is  equal  to  the  vertical  component  of  the  stress  in  the 
diagonal.     Hence 

T  2OOO 

Stress  in  9-10  =  — — —  =  12000(1.1547)  =  13856  Ibs. 

This  stress  is  a  pull,  as  may  be  seen  from  the  fact,  that,  in 
order  to  prevent  the  part  of  the  girder  to  the  right  of  the 
section  from  sliding  downwards  under  the  action  of  the  load, 
the  part  9-7  of  the  diagonal  9-10  must  pull  the  part  y-io  of 
the  same  diagonal. 

Next  take  moments  about  9 :  and,  since  the  moment  of  the 
stresses  in  9-1 1  and  9-10  about  9  is  zero,  we  must  have  that  the 
moment  of  the  stress  in  8-10;  i.e.,  the  product  of  this  stress 
by  the  height  of  the  girder,  must  equal  the  bending-moment. 


DETERMINING    THE  STRESSES  2N  A   BRIDGE-TRUSS.    189 


The  bending-moment  about  9  is 

8000  x  5  4-  4000  x  10  =  80000  foot-lbs. 
80000 


Hence 


Stress  in  8-10 


4-33 


80000(0.23094)  =  18475 


Proceed  in  a  similar  way  for  all  the  other  members.  The 
work  may  be  arranged  as  in  the  following  table ;  the  diagonal 
stresses  being  deduced  from  the  shearing-forces  by  multiplying 
by  1.1547,  and  the  chord  stresses  from  the  bending-moments 
by  multiplying  by  0.23094. 


2_ 

Stresses  in  Diagonals  cut 

Stresses  in  Chords  opposite  the 

JJ 

Shearing- 

by  Section,  in  Ibs. 

Bending- 

respective  Joints. 

c  .S> 

Force 

Moment,  in 

O     *•« 

in  Ibs. 

foot-lbs. 

*J 

Tension. 

Compression. 

Tension. 

Compression. 

I 

44OOO 

50806 

72OOOO 

166277 

2 

44OOO 

50806 

6lOOOO 

140873 

3 

36000 

4^69 

500000 

11547° 

4 

36OOO 

41569 

4IOOOO 

94685 

5 

28000 

32331 

\   320000 

73901 

6 

28OOO 

32331    \    250000 

57735 

7 

2OOOO 

23094 

'•    ISOOOO 

41569 

8 

20000 

23094 

I3OOOO 

30022 

9          i  2000 

13856 

80000 

18475 

10 

I2OOO 

13856 

5OOOO 

,"547 

II 

4OOO 

4619 

2OOOO 

4618 

12 

4OOO 

4619 

IOOOO 

2309 

EXAMPLE  II.  —  Given  the  truss  (Fig.  109)  loaded  at  each  oi 
the  lower  joints  with  10000  Ibs. :  find  the  stresses  in  the  members. 
The  length  of  chord  is  equal  to  the  length  of  diagonal  =  10  ft. 

Throughout  this  chapter,  tensions  will  be  written  with  the 
minus,  and  compressions  with  the  plus  sign. 

Solution. — Total  load  =  14(10000)  =  140000  Ibs. 

Each  supporting  force     =     70000    " 
The  entire  work  is  shown  in  the  following  tables:— 


i  go 


APPLIED   MECHANICS. 


CO    ^O       O\ 


II        II 


o     o     o 

to     >-o     to 

*t    VO     00 


CO      •«*•      CO      rt      CO      ^      CO 


x    x 


to  o  ^o  O  to  O 

••H  M  **  M  <->  0 

o    4-  +  +  +  +  + 

to  o  to  o  to  O 


I        I        I        I        I        I        I        I        I        I        I        I        I 
O      to     o      to     O 

X     X     X     X     X     X     X 


10      O       to 


>-o     O      to     O 
N      co     co     Tj- 


X     X     X     X     X     X 


888 

N       CO      CO 


1       1       1       1       1 


CO.     CO      N        M        >->        *- 

II         II         II         II         II         II         II         II 

I  I  I  I  I  I  I  I 


tt 

•N 


ONOO      t>sVO      torfcoN      "-I      O      ONOO      txO 
MNNNNMNNMN««H-,H4 


i-«       M       CO 


00       ON     O       NH       N       CO 


DETERMINING    THE  STRESSES  IN  A    BRIDGE-TRUSS, 


Numbers  of  Diagonals. 

Stresses 

in  Diagonals,  in  Ibs. 

I-    2 

28-29 

—  70000  X 

I-I547    = 

—  80829 

2-  3 

27-28 

+  60000   X 

I.T547    = 

+  69282 

3-  4 

26-27 

—  60000  X 

I.I547    = 

—  69282 

4-  5 

25-26 

+  50000   X 

I.I547    = 

+  57735 

5-6 

24-25 

—  50000  X 

LI547    = 

-57735 

6-  7 

23-24 

+40000  X 

I.I547    = 

+46188 

7-  8 

22-23 

—  40000   X 

LI547    = 

-46188 

8-  9 

21-22 

+  30000  X 

LI547    = 

+  34641 

9-10 

2O-2I 

—  30000  X 

LI547    = 

—  34641 

IO-II 

I9-2O 

+  20000    X 

LI547    = 

+  23094 

11-12 

18-19 

—  20000    X 

I.I547    = 

-23094 

12-13 

I7-I8 

+  IOOOO    X 

LI547    = 

+  II547 

I3-H 

16-17 

—  i  oooo  x 

I.I547   = 

-H547 

14-15 

I5-I6 

+  0 

0 

LOWER   CHORDS. 


IM  umbers  of  Chords. 

Stresses  in  Chords,  in  Ibs. 

2-  4 

26-28 

—      65OOOO 

X   0.11547  = 

-    75°56 

4-  6 

24-26 

—    I2OOOOO 

X   0.11547  = 

—  138564 

6-  8 

22-24 

—    I65OOOO 

X   0.11547  = 

—  190526 

8-10 

20-22 

—   2OOOOOO 

X   0.11547  = 

-230940 

10-12 

I  8-20 

—   225OOOO 

X   0.11547  = 

—  259808 

12-14 

16-18 

—    245OOOO 

X   0.11547  = 

-277128 

I4-l6 

—    245OOOO 

X  0.11547  = 

—  282902 

1 92 


APPLIED   MECHANICS. 


UPPER    CHORDS. 


Numbers  of  Chords. 

Stresses  in  Chords,  in  Ibs. 

'-    3 

27-29 

350000 

X  0.11547  = 

+  40415 

3-  5 

25-27 

950000 

x  0.11547  = 

+  109697 

5-  7 

23-25 

I45OOOO 

X  0.11547  = 

+  167432 

7-  9 

21-23 

1850000 

X   0.11547  = 

+  213620 

9-1  1 

19-21 

2I5OOOO 

X   0.11547  = 

+248261 

ii    13 

17-19 

2350000 

X   0.11547  = 

+  267355 

'j  -'5 

i5-i7 

245OOOO 

X   0.11547  = 

+  282902 

EXAMPLE  III.  —  Given  the  same  truss  as  in  Example  II., 
loaded  at  2,  4,  6,  8,  10,  and  12  with  10000  Ibs.  at  each  point, 
the  remaining  lower  joints  being  loaded  with  50000  Ibs.  at  each 
joint :  find  the  stresses  in  the  members. 

EXAMPLE  IV.  —  Given  a  semi-girder,  free  at  one  end  (Fig. 
112),  loaded  at  2,  4,  and  6  with  10000  Ibs.,  and  at  8,  10,  and  12 
with  5000  Ibs. :  find  the  stresses  in  the  members. 

TRAVELLING-LOAD. 

§148.  Half-Lattice  Girder:  Travelling-Load.  —  When  a 
girder  is  used  for  a  bridge,  it  is  not  subjected  all  the  time  to 
the  same  set  of  loads. 

The  load  in  this  case  consists  of  two  parts,  —  one,  the  dead 
load,  including  the  bridge  weight,  together  with  any  permanent 
load  that  may  rest  upon  the  bridge  ;  and  the  other,  the  moving 
or  variable  load,  also  called  the  travelling-load,  such  as  the 
weight  of  the  whole  or  part  of  a  railroad  train  if  it  is  a  railroad 
bridge,  or  the  weight  of  the  passing  teams,  etc.,  if  it  is  a  common- 
road  bridge.  Hence  it  is  necessary  that  we  should  be  able  to 
determine  the  amount  and  distribution  of  the  loads  upon  the 
bridge  which  will  produce  the  greatest  tension  or  the  greatest 


GREATEST  DIAGONAL   STRESSES  IN  GIRDER.  193 

compression  in  every  member,  and  the  consequent  stress  pro- 
duced. 

§149.  Greatest  Stresses  in  Semi-Girder.  —  Wherever  the 
section  be  assumed  in  a  semi-girder,  it  is  evident  that  any-  load 
placed  on  the  truss  at  any  point  between  the  section  and  the 
free  end  increases  both  the  shearing-force  and  the  bending- 
momerit  at  that  section,  and  that  any  load  placed  between  the 
section  and  the  fixed  end  has  no  effect  whatever  on  either 
the  shearing-force  or  the  bending-moment  at  that  section. 

Hence  every  member  of  a  semi-girder  will  have  a  greater 
stress  upon  it  when  the  entire  load  is  on,  than  with  any  partial 
load. 

§  150.  Greatest  Chord  Stresses  in  Girder  supported  at 
Both  Ends.  —  Every  load  which  is  placed  upon  the  truss,  no 
matter  where  it  is  placed,  will  produce  at  any  section  whatever  a 
bending-moment  tending  to  turn  the  two  parts  of  the  truss  on 
the  two  sides  of  the  section  upwards  from  the  supports  ;  i.e.,  so 
as  to  render  the  truss  concave  upwards. 

Hence  every  load  that  is  placed  upon  the  truss  causes  com- 
pression in  every  horizontal  upper  chord,  and  tension  in  every 
horizontal  lower  chord.  Hence,  in  order  to  obtain  the  greatest 
chord  stresses,  we  assume  the  whole  of  the  moving  load  to  be 
upon  the  bridge. 

§  151.  Greatest  Diagonal  Stresses  in  Girder  supported 
at  Both  Ends.  —  To  determine  the  distribution  of  the  load 
that  will  produce  the  greatest  stress  of  a  certain  kind  (tension 
or  compression)  in  any  given  diagonal,  let  us  suppose  the  diag- 
onal in  question  to  be  7-8  (Fig.  109),  through  which  we  take 
our  section  ab.  Now  it  is  evident  that  any  load  placed  on  the 
truss  between  ab  and  the  left-hand  (nearer)  support  will  cause  a 
shearing-force  at  that  section  which  will  tend  to  slide  the  part 
of  the  girder  to  the  left  of  the  section  downwards  with  refer- 
ence to -the  other  part,  and  hence  will  cause  a  compressive 
stress  in  7-8  ;  while  any  load  between  the  section  and  the  right- 


194  APPLIED  MECHANICS. 

hand  (farther)  support  will  cause  a  shearing-force  of  the  oppo- 
site kind,  and  hence  a  tension  in  the  bar  7-8. 

Now,  the  bridge  weight  itself  brings  an  equal  load  upon  each 
joint ;  hence,  when  the  bridge  weight  is  the  only  load  upon  the 
truss,  the  bar  7-8  is  in  tension. 

Hence,  any  load  placed  upon  the  truss  between  the  section 
and  the  farther  support  tends  to  increase  the  shearing-force  at 
that  section  due  to  the  dead  load  (provided  this  is  equally  dis- 
tributed among  the  joints) ;  whereas  any  load  placed  between 
the  section  and  the  nearer  support  tends  to  decrease  the  shear- 
ing-force at  the  section  due  to  the  dead  load,  or  to  produce  a 
shearing-force  of  the  opposite  kind  to  that  produced  by  the  dead 
load  at  that  section. 

Hence,  if  we  assume  the  dead  load  to  be  equally  distributed 
among  the  joints,  we  shall  have  the  two  following  propositions 
true  :  — 

(a)  In  order  to  determine  the  greatest  stress  in  any  diagonal 
which  is  of  the  same  kind  as  that  produced  by  the  dead  load, 
we  must  assume  the  moving  load  to  cover  all  the  panel  points 
between  the  section  and  the  farther  abutment,  and  no  other 
panel  points. 

(b)  In  order  to  determine  the  greatest  stress  in  any  diagonal 
of  the  opposite  kind  to  that  produced  by  the  dead  load,  we  must 
assume  the  moving  load  to  cover  all  the  panel  points  between 
the  section  and  the  nearer  abutment,  and  no  others. 

This  will  be  made  clear  by  an  example. 

EXAMPLE  I.  —  Given  the  truss  shown  in  Fig.  113.     Length 

of  chord  =  length  of  diagonal  = 
A   A    A   g  !Lu    10  feet.     Dead  load  =  8000  Ibs. 


Y  4   Y  Y  Y  Y  Y  Ypl    applied  at  each  upper  panel  point. 
FIG  ™    Moving  load  =  30000  Ibs.  applied 

at  each  upper  panel  point.     Find 
the  greatest  stresses  in  the  members. 


EXAMPLE   OF  BRIDGE-TRUSS, 


195 


Solution,     (a)   Chord  Stresses. — Assume  the  whole  load  to 
be    upon   the   bridge : 
this    will    give    38000 
each 


(1)  +      76788  (3)    4-.  20R423  (5)  -f-  296181   (7;  +  340059  (9) 


(2)    -153575  (4)     -263272(6)   -329090  (8)-  3510251 


Ibs.  at  eacn  upper 
panel  point ;  i.e.,  omit- 
ting I  and  17,  where 
the  load  acts  directly 
on  the  support,  and 
not  on  the  truss.  FlG'  II4- 

Hence,  considering  the  bridge  so  loaded,  we  shall  have  the  fol- 
lowing results  for  the  chord  stresses  :  — 

Each  supporting  force  =  sSooof-J  —  133000. 


Section  at 

Bending-Moment,  in  foot-lbs. 

2       16 

133000  x  5 

=  665000 

3    15 

133000  X  IO 

=  1330000 

4    14 

133000  X  15 

—  38000  x  5 

=  1805000 

5    13 

133000  X  2O 

—  38000  X  10 

=  2280000 

6      12 

133000  X  25 

-  38ooo(  5  -f-  15) 

=  2565000 

7    ii 

133000  X  30 

-  38000(10  +  20) 

=  2850000 

8    10 

133000  x  35 

—  38ooo(  5  +  15  -{- 

25)  =  2945000 

9 

133000  X  40 

—  38000(10  4-  20  -j- 

30)  =  3040000 

Numbers  of  Chords. 

Stresses  in  Upper 

Chords. 

i-3 

I5-I7 

665000   X   0.11547 

=    +     76788 

3-5 

I3-I5 

1805000   X   0.11547 

=    +208423 

5-7 

11-13 

2565000   X   0.11547 

=    +296181 

7-9 

9-1  1 

2945000   X   0.11547 

=    +340059 

APPLIED   MECHANICS. 


Numbers  of  Chords. 

Stresses 

in  Lower  Chords. 

2-  4 

14-16 

—  1330000  X 

O.II547   = 

-153575 

4-  6 

12-14 

—  2280000  X 

O.II547   = 

-263272 

6-  8 

IO-I2 

—  2850000   X 

O.II547    = 

—  329090 

8-10 

—  3040000  X 

O.II547    = 

-351029 

Next,  as  to  the  diagonals,  take,  for  instance,  the  diagonal 
7-8.  When  the  dead  load  alone  is  on  the  bridge,  the  diagonal 
7-8  is  in  tension.  From  the  preceding,  we  see  that  the  greatest 
tension  is  produced  in  this  bar  when  the  moving  load  is  on  the 
points  9,  n,  13,  and  15,  and  the  dead  load  only  on  the  points  3, 
5,  7.  Now,  a  load  of  38000  Ibs.  at  13,  for  instance,  causes  a 

shearing-force  of  —(38000)  =  9500  Ibs.  at  any  section  to  the 

10 

left  of  13;  and  this  shearing-force  tends  to  cause  the  part  to 
the  left  of  the  section  to  slide  upwards,  and  that  to  the  right 
downwards. 

On  the  other  hand,  with  the  same  load  at  the  same  place, 

there  is  produced  a  shearing-force  of  —  (38000)  =  28500    Ibs. 

16 

at  any  section  to  the  right  of  13  ;  and  this  shearing-force  tends 
to  cause  the  part  to  the  left  to  slide  downwards,  and  that  to  the 
right  upwards.  Paying  attention  to  this  fact,  we  shall  have, 
when  the  loads  are  distributed  as  above  described,  a  shearing- 
force  at  the  bar  7-8  causing  tension  in  this  bar  ;  the  magnitude 
of  this  shearing-force  being 


6  +  8)  _ 


i6  16 

Hence,  we  may  arrange  the  work  as  follows  :  — 


6)  =  41500. 


GREATEST  DIAGONAL    STRESSES  IN  GIRDER. 


197 


Greatest 

Stresses  in 

Numbers  of 

Greatest  Shearing-Forces  producing  Stresses  of  Same  Kind  as 

Diagonals  of 

Diagonals. 

Dead  Load. 

as  those  due 

to  Dead 

Load. 

1-2 

17-16 

3^(2  +  4+6+8+IO+I2+I4)                              =   133000 

-'53575 

2-3 

16-15 

^(2+4+6+8+10+12+14)                       =  I33ooo 

+  '53575 

3-4 

'5-H 

3^(2+4+6+8+io+I2)-~(2)              =    98750 

—  114027 

4-5 

14-13 

^(2+4+6+8+10+12)  -^(2)              =    98750 

+  i  14027 

5-6 

13-12 

^(2+4+6+8+10)          -^(2+4)        =    68250 

-  78808 

6-7 

I  2-1  1 

^(2+4+6+8  +  10)          -^(2+4)        -    68250 

+  78808 

7-8 

II-IO 

^(2+4+6+8)                 -^(2+4+6)  =    41500 

-  47920 

8-9 

io-  9 

^(2+4+6+8)                 -  ^(2+4+6)  -    41500 

+  47920 

Greatest 

Stresses  in 

Numbers  of 

Greatest  Shearing-Forces  producing  Stresses  of  Kind  Opposite 

Diagonals  ^pf 

Diagonals. 

from  Dead  Load. 

Kind  Oppo- 

site from 

Dead  Load. 

8-9 

io-  9 

^°(2+4+6)  -  ^(2+4+6+8)                   -  18500 

-21362 

7-8 

II-IO 

^(2+4+6)  -  ?^(2+4+6+8)                   -  18500 

+  21362 

The  diagonals  7-8,  8-9,  9-10,  and  10-11  are  the  only  ones 
that,  under  any  circumstances,  can  have  a  stress  of  the  kind 
opposite  to  that  to  which  they  are  subjected  under  the  dead 
load  alone. 


I98  APPLIED   MECHANICS. 

Fig.  114  exhibits  the  manner  of  writing  the  stresses  on  the 
diagram. 

§  152.  General  Application  of  this  Method.  —  It  is  plain 
that  the  method  used  above  will  apply  to  any  single  system  of 
bridge-truss  with  horizontal  chords  and  diagonal  bracing,  what- 
ever be  the  inclination  of  the  braces. 

When  seeking  the  stress  in  a  diagonal,  the  section  must  be 
so  taken  as  to  cut  that  diagonal ;  and,  as  far  as  this  stress  alone 
is  concerned,  it  may  be  equally  well  taken  at  any  point,  as  well 
as  near  a  joint,  provided  only  it  cuts  that  diagonal  which  is  in 
action  under  the  load  that  produces  the  greatest  stress  in  this 
one,  and  no  other. 

On  the  other  hand,  when  we  seek  the  stress  in  a  horizontal 
chord,  the  section  might  very  properly  be  taken  through  the 
joint  opposite  that  chord. 

Taking  it  very  near  the  joint,  only  serves  to  make  one  sec- 
tion answer  both  purposes  simultaneously. 

§  153.  Bridge-Trusses  with  Vertical  and  Diagonal  Bra- 
cing.—  When,  as  in  Figs,  in  and  112,  there  are  both  vertical 
and  diagonal  braces,  and  also  horizontal  chords,  we  may  deter- 
mine the  stresses  in  the  diagonals  and  in  the  chords  just  as 
before  ;  only  we  must  take  the  section  just  to  one  side  of  a  joint, 
and  never  through  the  joint. 

As  to  the  verticals,  in  order  to  determine  the  stress  in  any 
vertical,  we  must  impose  the  conditions  of  equilibrium  between 
the  vertical  components  of  the  forces  acting  at  one  end  of  that 
vertical:  thus,  if  the  loads  are  at  the  upper  joints  in  Fig.  in, 
then  the  stress  in  vertical  3-2  must  be  equal  and  opposite  to 
the  vertical  component  of  the  stress  in  diagonal  1-2,  as  these 
stresses  are  the  only  vertical  forces  acting  at  joint  2. 

Vertical  5-4  has  for  its  stress  the  vertical  component  of  the 
stress  in  3-4,  etc.  Thus 

Stress  in  3-2  =  shearing-force  in  panel  1-3, 
Stress  in  5-4  =  shearing-force  in  panel  3-5,  etc.- 


TRUSSES    WITH    VERTICAL   AND  DIAGONAL   BRACING.    199 

On  the  other  hand,  if  the  loads  be  applied  at  the  lower 
joints,  then 

Stress  in  3-2  =  shearing-force  in  panel  3-5, 
Stress  in  5-4  =  shearing-force  in  panel  5-7,  etc. 

EXAMPLE.  —  Given  the  truss  shown  in  Fig.  in.  Given 
panel  length  =  height  of  truss  —  10  feet,  dead  load  per  panel 
point  =  12000  Ibs.,  moving  load  per  panel  point  =  23000  Ibs.  ; 
load  applied  at  upper  joints. 

Solution,  (a)  Chord  Stresses.  —  Assume  the  entire  load  on 
the  bridge,  i.e.,  35000  Ibs.  per  panel  point.  Hence 

Total  load  on  truss  =13  (35000)  =  455000  Ibs., 
Each  supporting  force  =  227500  Ibs. 


Joint  near 
which 
Section  is 
taken. 

Bending-Moment  at  the  Section  very  near  the  Joint,  on 

Either  Side  of  the  Joint. 

I        28 

0 

3      27 

227500  x  10 

=  2275000 

5      25 

227500  X   20  —  35000  X   10 

=  4200000 

7      23 

227500  X  30  —  35000(10  +  20) 

=  5775000 

9     21 

227500  X  40  —  35000  (  10  +  20  +  30) 

—  7000000 

ir      19 

227500  X   50  —  35000  (10  +  20  +  30  +  40) 

=  7875000 

13      17 

227500  X  60  —  35000(10  +  20  -h  30  +  40  + 

50)              =  8400000 

IS 

227500  X   70  —  35000  (10  +  20  +  30  +  40  + 

50  -f-  60)  =  8575000 

To  find  any  chord  stress,  divide  the  bending-moment  at  a 
section  cutting  the  chord,  and  passing  close  to  the  opposite 
joint,  by  the  height  of  the  girder,  which  in  this  case  is  10. 
Hence  we  have  for  the  chord  stresses  (denoting,  as  before,  com- 
pression by  +,  and  tension  by  — ) :  — 


2OO 


APPLIED   MECHANICS. 


Stresses  in  Upper  Chords. 

Stresses  in  Lower  Chords. 

i-  3 

27-28 

+  227500 

2-  4 

24-26 

—  227500 

3-  5 

25-27 

4-420000 

4-  6 

22-24 

—  420000 

5-  7 

23-25 

+  5775°° 

6-  8 

20-22 

-5775°° 

7-  9 

21-23 

+  700000 

8-10 

18-20 

—  700000 

9—1  1 

19-21 

+  787500 

IO-I2 

1  6-1  8 

-787500 

11-13 

17-19 

+  840000 

12-14 

14-16 

—  840000 

i3-!5 

iS-1? 

+  8575°° 

Diagonals.  —  It  is  evident,  that,  for  the  diagonals,  the  same 
rule  holds  as  in  the  case  of  the  Warren  girder  :  i.e.,  the  greatest 
stress  of  the  same  kind  as  that  produced  by  the  dead  load 
occurs  when  the  moving  load  is  on  all  the  joints  between  the 
diagonal  in  question  and  the  farther  abutment ;  whereas  the 
greatest  stress  of  the  opposite  kind  occurs  when  the  moving 
load  covers  all  the  joints  between  the  diagonal  in  question  and 
the  nearer  abutment. 

The  work  of  determining  the  greatest  shearing-forces  may 
be  arranged  as  in  tables  on  p.  191. 

Counterbraces.  —  If  the  truss  were  constructed  with  those 
diagonals  only  that  slope  downwards  towards  the  centre,  and 
which  may  be  called  the  main  braces,  the  diagonals  1 1-12, 
13-14,  14-17,  and  16-19  would  sometimes  be  called  upon  to 
bear  a  thrust,  and  the  verticals  12-13  and  17-16  a  pull  :  this 
would  necessitate  making  these  diagonals  sufficiently  strong 
to  resist  the  greatest  thrust  to  which  they  are  liable,  and  fixing 
the  verticals  in  such  a  way  as  to  enable  them  to  bear  a  pull. 

In  order  to  avoid  this,  the  diagonals  10-13,  12-15,  I5~I6, 
and  17-18  are  inserted,  which  are  called  counterbraces,  and 
which  come  into  action  only  when  the  corresponding  main 


TRUSSES    WITH    VERTICAL    AND   DIAGONAL   BRACING.  2OI 


braces    would   otherwise   be   subjected    to   thrust.     They  also 
prevent  any  tension  in  the  verticals. 


Diagonals. 

Greatest  Shearing-  Forces  of  the  Same  Kind  as  those  produced  by 
Dead  Load. 

I-   2 

28-26 

^I+2  +  3+...+I3) 

=  227500 

3-  4 

27-24 

3J^(l  +  2  +  3+...  +  I2)_I5^(l) 

=   I94H3 

5-6 

25-22 

3-^P(l  +  2  +  3+  ...  +  II)-  '-^(1  +  2) 

=   162429 

7-8 

23-20 

^(i  +  2+3+  .  .  .  +  10)  -  x-^?(i  +  2+3) 

J-4                                                                                                                         J-4 

=   132357 

9-10 

2I-I8 

^(1  +  2+3+  ...+  9)-^p(i  +  2+.. 

+  4)  =  103929 

11-12 

19-16 

33222(1  +  2+3+...+  8)-^°(i  +  2+.. 

+  5)=    77H3 

I3T4 

17-14 

3^(l  +  2  +  3+...+    7)_^I  +  2+.. 

+  6)=    52000 

Diagonals. 

Greatest  Shearing-Forces  of  the  Opposite  Kind  to  those  produced  by 
Dead  Load. 

•3-14 

17-14 

^(I+2+3+...  +  6)_^I+2+...+7) 

=          28500 

11-12 

19-16 

22f(i  +  2+        ...  +  5)_H5p(r  +  2  +  ...  +  8) 

=             6643 

9-10 

2I-I8 

^(,  +  2+        ...  +  4)-^(l  +  2  +  ...  +  9) 

=  ~i357i 

The  main  braces  and  counterbraces  of  a  panel  are  never  in 
action  simultaneously.  Hence  we  have,  for  the  greatest  stresses 
in  the  diagonals,  the  following  results,  obtained  by  multiplying 

the  corresponding  shearing-forces  by — -  —  1.414. 

cos  45 


2O2 


APPLIED   MECHANICS. 


In  the  following  I  have  used  this  number  to  three  decimal 
places,  as  being  sufficiently  accurate  for  practical  purposes. 


Stresses  in  Main  Braces. 

Stresses  in  Counterbraces. 

I-    2 

28-26 

-321685 

15-12 

15-16 

—  40299 

3-  4 

27-24 

-274518 

I3-IO 

I7-I8 

-  9393 

5-6 

25-22 

-229675 

7-  8 

23-20 

-187153 

9-10 

21-18 

—  146956 

11-12 

19-16 

—  109080 

i3~I4 

17-14 

-   73528 

Vertical  Posts.  —  Since  the  loads  are  applied  at  the  upper 
joints,  the  conditions  of  equilibrium  at  the  lower  joints  require 
that  the  thrust  in  any  vertical  post  shall  be  equal  to  the  vertical 
component  of  the  tension  in  that  diagonal  which,  being  in  action 
at  the  time,  meets  it  at  its  lower  end. 

Hence  it  is  equal  to  the  shearing-force  in  that  panel  where 
the  acting  diagonal  meets  it  at  its  lower  end. 

We  therefore  have,  for  the  posts,  the  following  as  the  greatest 
thrusts : — 

STRESSES   IN   VERTICALS. 


3-  2 

27-26 

+  2275OO 

5-  4 

25-24 

+  I94M3 

7-  6 

23-22 

+  162429 

9-  8 

2I-2O 

+  132357 

II-IO 

I9-I8 

+  103929 

13-12 

I7-l6 

+     77143 

i5-J4 

+     52000 

CONCENTRATING    THE   LOAD   AT  THE  JOINTS. 


203 


X 


X 


X 


FIG. 


Fig.  115  shows  the  stresses  marked  on  the  diagram. 

§  154.  Manner  of  Concentrating  the  Load  at  the  Joints. 

—  In  using  the  methods  given  above,  we  are 

assuming  that  all  the  loads  are  concentrated 

at  the  joints,  and  that  none  are  distributed 

over  any  of  the  pieces.     As  far  as  the  mov- 
ing load  is  concerned,  and   also  all  of   the 

dead   load    except    the  weight   of   the  truss 

itself,  this  always  is,  or  ought  to  be,  effected ; 

and  it  is  accomplished  in  a  manner  similar 

to  that  adopted  in  the  case  of  roof-trusses. 

This  method   is   shown   in  the  figure   (Fig. 

1 1 6);  floor-beams  being  laid  across  from 
girder  to  girder  at  the  joints, 
on  top  of  which  are  laid  longi- 
tudinal beams,  and  on  these 
the  sleepers  if  it  is  a  railroad 
bridge,  or  the  floor  if  it  is  a 
road  bridge.  The  weight  of 
the  truss  itself  is  so  small  a 
part  of  what  the  bridge  is 
called  upon  to  bear,  that  it 
can,  without  appreciable  error, 
be  considered  as  concentrated 
at  the  joints  either  of  the  up- 
per chord,  of  the  lower  chord, 
or  of  both,  according  to  the 
manner  in  which  the  rest  of 
the  load  is  distributed. 

§  155.  Closer  Approxima- 
tion to  Actual  Shearing- 
Force. —  In  our  computations 
of  greatest  shearing-force,  we 


FIG.  115. 


make  an  approximation  which   is  generally  considered  to  be 


APPLIED   MECHANICS. 


sufficiently  close,  and  which  is  always  on  the  safe  side.  To 
illustrate  it,  take  the  case  of  panel  3-5  of  the  last  example. 
In  determining  its  greatest  shearing-force,  we  considered  a  load 
of  35000  Ibs.  per  panel  point  to  rest  on  all  the  joints  from  the 
right-hand  support  to  joint  5,  inclusive,  and  the  dead  load  to 
rest  on  all  the  other  joints  of  the~truss.  Now,  it  is  impossible, 
if  the  load  is  distributed  uniformly  on  the  floor  of  the  bridge, 
to  have  a  load  of  35000  Ibs.  on  5  and  12000  on  3  simultaneously  ; 
for,  if  the  moving  load  extended  on  the  bridge  floor  only  up  to 
5,  the  load  on  5  would  be  only  12000  +  ^-(23000)  =  23500  Ibs., 
and  that  on  3  would  then  be  12000  Ibs.  If,  on  the  other  hand, 
the  moving  load  extends  beyond  5  at  all,  as  it  must  if  the  load 
on  5  is  to  be  greater  than  23500  Ibs.,  then  part  of  it  will  rest 
on  3,  and  the  load  on  3  will  then  be  greater  than  12000  Ibs.  ; 
for  whatever  load  there  is  between  3  and  5  is  supported  at 
3  and  5. 

Moreover,  we  know  that  the  effect  of  increasing  the  load  on 
5  is  to  increase  the  shearing-force,  provided  we  do  not  at  the 
same  time  increase  that  on  3  so  much  as  to  destroy  the  effect 
of  increasing  that  on  5. 

Hence,  there  must  be  some  point  between  3  and  5  to  which 
the  moving  load  must  extend  in  order  to  render  the  shearing- 
force  in  panel  3-5  a  maximum. 

Let  the  distance  of  this  point  from  5  be^r;  then,  if  we  let 


w  — 


=  moving  load  per  foot  of  length, 

Moving  load  on  panel  =  wx, 

Part  supported  at  3       =  --  , 

20 


Part  supported  at  5       =  wx  —       -. 

20 

Hence,  portion  of  shearing-force  due  to  the  moving  load  on 
panel  3-5  equals 


CONCENTRATING    THE  LOAD   AT  THE  JOINTS. 


I2/  WX2\  I     WX2          W  I  I*X2\ 

—  (  WX   —    --  )   ---   =   —  (  I2X    --  -  —  ). 

i4\  20  /        14   20         i4\  20  / 

This  becomes  a  maximum  when  its  first  differential  co-efficient 
becomes  zero,  i.e.,  when 


therefore 


12  -      x  =  o 


X      =    9.23. 


Hence,  when  the  moving  load  extends  to  a  distance  of  9.23  feet 
from  5,  then  the  shearing-force  in  panel  3-5,  and  hence  the 
stress  in  diagonal  3-4,  is  a  maximum. 


Panels. 

Portion  of  Shearing-Force 
due  to  Moving  Load  on 
Panel. 

Value 
of  xt  in 
feet. 

Portion  of  Load 
at  Joints  named 
below. 

Portion  of  Load 
at  Joints  named 
below. 

i-  3 

27-28 

iv  (             I3*A 

IO.OO 

« 

11500 

3 

11500 

i4\  J       20  y 

3-  5 

5-  7 

25-27 
23-25 

I4\                     20   / 

9-23 
8.46 

3 

5 

9797 
8230 

5 
7 

11432 
11227 

I4\                     20   / 

7-  9 

21-23 

I4\                     20   / 

7.69 

7 

6801 

9 

10886 

9-1  1 

19-21 

H(  9X  ~  ^f?) 

6.92 

9 

5507 

n 

10409 

11-13 
13-15 

17-19 
15-17 

14  \                     20  / 

u<l  ^          I3^"2\ 

6.15 
5.38 

ii 
13 

4350 
3329 

13 
15 

9795 
9045   1 

I4\    ''               20   / 

To  show  how  the  adoption  of  this  method  would  affect  the 
resulting  stresses  in  the  diagonals  and  verticals,  I  have  given 
the  work  above,  and  shown  the  difference  between  these  and 


206 


APPLIED   MECHANICS. 


the  former  results.     In  this  table  x  =  distance  covered  by  load 
from  end  of  panel  nearest  the  centre. 


Panels. 


Greatest  Shearing-Force  of  Same  Kind  as  that  due  to  Dead  Load. 


3-  5 
5-  7 
7-  9 


11-13 


27-28 
25-27 
23-25 

2I-23 

19-21 
17-19 
15-17 


3500Q, 
14    l 


—  227500 

=  193385 

=  161038 


-101654 


— (i+...+S)=  49345 


Hence,  for  stresses  in  main  braces,  we  have 


Diagonals. 

Stresses. 

I-    2 

28-26 

-321685 

3-  4 

27-24 

-273446 

5-  6 

25-22 

—  227708 

7-  8 

23-20 

-184472 

9-10 

21-18 

—  143739 

11-12 

19-16 

—  105507 

i3-J4 

17-14 

—    69774 

Moreover,  for  the  shearing-forces   of   opposite   kind   from 


CONCENTRATING    THE  LOAD   AT   THE  JOINTS.          2O/ 


those  due  to  dead  load,  we  have,  if  x  =  distance  from  end  of 
panel  nearest  support  which  is  covered  by  moving  load,  — 


Panels. 

Portion  of  Shear  due 
to  Moving  Load  on  Panel. 

Value 
otx. 

Portion  of  Load 
at  Joints  named 
below. 

Portion  of  Load 
at  Joints  named 
below. 

17-15 

«/6*  -  ^} 

4.62 

15 

2455 

13 

8171 

I4\              20  / 

11-13 

19-17 

E(SX  -  l^f] 

3.84 

13 

1695 

II 

7137 

I4\              20  / 

Panels. 

Greatest  Shearing-Forces  of  Opposite  Kind  from  those  due  to  Dead 

Load. 

13-5 

17-15 

S22(i+...+ 

5)  +  l(3:67I)-f4,I4455,-"i7(l+...+6)  = 

25846 

M-.3 

1^-17 

35000  <i  -j-      + 

4)+l(3o637)_f4(l3695)_^(l+...+7)  = 

4116 

Hence  we  have  the  following  as  the  stresses  in  the  counter- 
braces  :  — 


Counter-Braces. 

Stresses. 

15-12 
13-10 

15-16 
I7-l8 

-    36546 

—        5820 

And,  for  the  verticals,  we  have  the  new,  instead  of  the  old, 
shearing-forces. 


208 


APPLIED  MECHANICS. 


The  following  table  compares  the  results  :  — 


Diagonals. 

Stress,  Ordinary 
Method. 

Stress,  New  Method. 

Difference. 

I-    2 

28-26 

-321685 

-321685 

3-  4 

27-24 

-274518 

-273446 

1072 

5-  6 

25-22 

-229675 

-227708 

1967 

7-  8 

23-20 

-187153 

-184472 

268l 

9-10 

21-18 

-146956 

-J43739 

3217 

11-12 

19-16 

—  109080 

—  105507 

3573 

i3-J4 

17-14 

-     73528 

-    69774 

3754 

15—12 

15-16 

-    40299 

—  36546 

3753 

13-10 

17-18 

-     9393 

—   5820 

3573 

Verticals. 

Stress,  Ordinary 
Method. 

Stress,  New  Method. 

Difference. 

3-  2 

27-26 

-f227500 

+  227500 

O 

5-  4 

25-24 

+  I94H3 

+  193385 

758 

7-  6 

23-22 

4-162429 

+  161038 

I391 

9-  8 

21-20 

+  T32357 

+  130461 

1896 

II-IO 

I9-I8 

4-103929 

+  101654 

2275 

13-12 

I7-l6 

+   77H3 

+    74616 

2527 

i5-J4 

4-'  28500 

+  49345 

2655 

§156.   Compound  Bridge-Trusses The  trusses  already 

discussed  have  contained  but  a  single  system  of  latticing,  or 


COMPO  UND   BRID  GE-  TR  USSES. 


209 


at  least  only  one  system  that  comes  in  play  at  one  time ;  so  that 
a  vertical  section  never  cuts  more  than  three  bars  that  are  in 
action  simultaneously,  the  main  brace  having  no  stress  upon  it 
when  the  counterbrace  is  in  action,  and  vice  versa. 

We  may,  however,  have  bridge-trusses  with  more  than-  one 
system  of  lattices ;  and,  in  determining  the  stresses  in  their 
members,  we  must  resolve  them  into  their  component  systems, 
and  determine  the  greatest  stress  in  each  system  separately, 
and  then,  for  bars  which  are  common  to  the  two  systems,  add 
together  the  stresses  brought  about  by  each. 

In  some  cases,  the  design  is  such  that  it  is  possible  to 
resolve  the  truss  into  systems  in  more  than  one  way,  and  then 
there  arises  an  uncertainty  as  to  which  course  the  stresses  will 
actually  pursue. 

In  such  cases,  the  only  safe  way  is  to  determine  the  greatest 
stress  in  each  piece  with  every  possible  mode  of  resolution  of 
the  systems,  and  then  to  design  each  piece  in  such  a  way  as  to 
be  able  to  resist  that  stress. 

Generally,  however,  such  ambiguity  is  an  indication  of  a 
waste  of  material ;  as  it  is  most  economical  to  put  in  the  bridge 
only  those  pieces  that  are  absolutely  necessary  to  bear  the 
stresses,  as  other  pieces  only  add  so  much  weight  to  the  struc- 
ture, and  are  useless  to  bear  the  load. 

The  mode  of  proceeding  can  be  best  explained  by  some 
examples. 

EXAMPLE  I.  —  Given  the  lattice-girder  shown  in  Fig.  117, 
loaded  at  the  lower  panel  points 

1  TA         i     i          i  11  1     3     5    7     9    11    13    13  17  19  21  23 

only.       Dead   load  =  7200  Ibs. 

per  panel    point,    moving   load 

—  18000  Ibs  per  panel   point; 

let  the  entire  length  of  bridge  FlG' "7* 

be  60  feet ;    let   the  angle   made   by  braces   with   horizontal 

=  60°. 


210 


APPLIED   MECHANICS. 


+75600 
FIG.  ujc. 


19         23 


10        14        18        22 


This  truss  evidently  consists  of  the  two  single  trusses  shown 
in     Figs.     \\ja 
njb; 


and  njb ;  and 
we  can  compute 
the  greatest 
stress  of  each 

kind  in  each  member  of  these  trusses,  and  thus 
obtain    at    once 
all    the    diag- 
onal    stresses, 
and     then,    by    E3  Ec±4 

....  FIG.  117*. 

addition,  the 
greatest  chord  stresses. 

Thus  the  stress  in  1-3  (Fig.  117)  is  the 
same  as  the  stress  in  1-5  (Fig.  I  ija). 

The  stress  in  3-5  =  stress  in  1-5  (Fig. 
1170)  +  stress  in  3-7  (Fig.  117^). 

The  stress  in  5-7  =  stress  in  5-9  (Fig. 
117*7)  -f  stress  in  3-7  (Fig.  117^). 

The  results  are  given  on  the  diagram  (Fig. 
117^);  the  work  being  left  for  the  student,  as 
it  is  similar  to  that  done  heretofore. 

EXAMPLE  II.  —  Given  the  lattice-girder 
shown  in  Fig.  1 1 8.  Given,  as  before,  Dead 
load  =  7200  Ibs.  per  panel  point,  moving  load 
=  18000  Ibs.  per  panel  point,  entire  length  of 
bridge  =  25  feet ;  load  applied  at  lower  panel 
points. 


Solution. —In  this  case,  there  are  two  possible  modes  of 
resolving  it  into  systems.  The  first  is  shown  in  Figs.  uSa  and 
n%b :  and  this  is  necessarily  the  mode  of  division  that  must 
hold  whenever  the  load  is  unevenly  distributed,  or  when  the 


COMPO  UND  BRID  GE-  TR  USSES. 


211 


travelling-load  covers  only  a  part  of  the  bridge ;  for  a  single 
load  at  6  is  necessarily  put  in  communication  with  the  support 
at  2  by  means  of  the  diagonals  6-3  and  3-2,  and  with  the  sup- 
port at  12  by  means  of  the  diagonals  6-7,  7-10,  lo-n,  and  the 
vertical  11-12,  and  can  cause  no  stress  in  the  other  diagonals 


1      3     5     7     9     11 


7  11 


6  10    12 

FIG.  ii&r. 


24  8  12 

FIG.  i i 83. 


5      7  11 


Z37I 


10    12 


FIG.  II&T. 


5     7 


When,  however,  the  whole  travelling-load  is  on  the  bridge, 
it  is  perfectly  possible  to  divide  it  into  the  two  trusses  shown 
in  Figs.  II&T  and  n&/,  the  diagonals  4-5,  7-10,  6-7,  and  5-8 
having  no  stress  upon  them. 

When  the  load  is  unevenly  distributed,  we  have  certainly 
the  first  method  of  division  ;  and  when  evenly,  we  are  not  sure 
which  will  hold. 

Hence  we  must  compute  the  greatest  stresses  with  each 
mode  of  division,  and  use  for  each  member  the  greatest ;  for 
thus  only  shall  we  be  sure  that  the  truss  is  made  strong 
enough. 

We  shall  thus  have  the  following  results :  — 


212 


APPLIED   MECHANICS. 


FIRST   MODE   OF   DIVISION    (FIGS.  n8«   AND 


Diagonals. 

Greatest  Shearing-Force 
of  One  Kind. 

Greatest  Shearing-Force 
of  Opposite  Kind. 

Corresponding 
Stresses. 

Fig. 
n8a. 

Fig. 
118*. 

2-  3 

12-9 

~~~(3  +  J)      =  20160 

O 

+23279 

—            0 

3-6 

9-3 

~(3+.i)      =  20160 

0 

-23279 

+        o 

6-  7 

8-5 

25200      7200, 
z  —  (2)  =    2160 

25200.,   .         7200           ..... 
(2)  =     0040 

+  2494 

—  9976 

7-10 
10-11 

5-4 
4-1 

25200      7200,  . 
--  —  —  -  —  (2)  =    2160 

0                                =             0 

25200,        ,                        nr»oin 

-   2494 

0 

+  9976 
—34918 

.    (2  +  4;     —  3024° 

Chords. 
Supporting  force  at  2  (Fig.  uSa)  or  12  (Fig. 

=  '-*?  (3  + 
Supporting  force  at  12  (Fig.  u8a)  or  2  (Fig. 


=  20160, 


Section. 

Chords. 

Maxi- 

i     mum 

Com- 

Bending-Moment. 

i  Stresses 
in 

Chords. 

ponents 
of 

Greatest 
Resultant 

«s 

S 

« 

•si 

i  "2 

00 

H     !  Separate 

Stresses. 

Stresses. 

M 

bib 

bi) 

bi      Trusses. 

j 

£ 

£ 

£ 

£    i 

1 

i  3 

9 

20i6oX   5  =  100800     2-  6 

8-12 

—  11639 

1-3 

9-1  1  i       o-f-  1-5 

+  17459 

6 

8 

20160X10=201600!   3-  7 

5-  9+23279 

3-5 

7-  9>  7+1-5 

+40738  : 

7 

5 

20160X15  —  25200 

X5  =  i?6400 

6-10 

4-  8  —20369 

5-7 

3-  7+5-9 

+46558 

10 

4 

30240X   5  =  151200 

7-1  1 

I-  5  +17459  2-4 

10-12  2-   6+2-4 

-11639 

10-12 

2-  4 

O 

4-6 

8-102-  6+4-8 

—  32008  ! 

6-8 

6-10+4-8 

-40738  ! 

COMPOUND   BRIDGE-TRUSSES. 


2I3 


SECOND    METHOD    OF    DIVISION    (FIGS.  nSc    AND 
Diagonals  (Fig.  n8<r). 


Diagonals. 

Maximum 
Shear. 

Corresponding 
Stresses. 

1-4 

10-11 

252OO 

—  29098 

4-5 

7-10 

0 

O 

Fig.  u&/. 


•    Diagonals. 

Maximum 
Shear. 

Corresponding 
Stresses. 

2-3 

9-12 

25200 

+  29098 

3-6 

8-9 

25200 

—  29098 

6-7 

5-8 

O 

O 

Chords. 

Each  supporting  force  in  either  figure  =  25200. 
Fig.  n8c. 

Bending-moment  anywhere  between  4  and  10  =  (25200)  (5)  =  126000; 

•  /.     Stress  in  i-n  =  +14549, 
.*.     Stress  in  4-10  =  —14549. 

Fig.  n8d. 

Bending-moment  at  3  or  9  =  126000, 

Bending-moment  anywhere  between  6  and  8  =  252000; 

/.  Stress  in  3-9  =  4-29098, 
Stress  in  2-6  or  8-12  =  —14549, 
Stress  in  6-8  =  —29098. 


214 


APPLIED   MECHANICS. 


Hence  we  have  for  chord  stresses,  with  this  second  divis- 
ion, — 


Chords. 

Stresses. 

i-3 

9-1  1 

I-II    -|-   0 

+  14549 

3-5 

7-  9 

i-n  +  3-9 

+  43647 

5-7 

-    . 

i-n  +  3-9 

+  43647 

2-4 

IO-I2 

0   -f-    2-6 

-14549 

4-6 

8-10 

4—10  4-  2-6 

—  29098 

6-8 

— 

4-10  +  6-8 

—  43647 

Hence,  selecting  for  each  bar  the  greatest,  we  shall  have,  as 
the  stresses  which  the  truss  must  be  able  to  resist,  — 


1-4 

IO-II 

+         o 

-34918 

i-3 

9-1  1 

+  17459 

2-3 

12-9 

+  29098 

0 

3-5 

7-  9 

+43647 

3-6 

9-8 

+         o 

—  29098 

5-7 

- 

+46558 

4-5 

10-  7 

+  9976 

—  2494 

2-4 

IO-I2 

-14549 

5-8 

7-  6 

+  2494 

-  9976 

4-6 

8-10 

—  32008 

6-8 

—43647 

These  results  are  recorded  in  Fiff.  uSe. 


(1)+17459  (3)+  43647(5)+  46558(7)+  43647(9^17459(11) 


)- 32008  (6)  -  43647  (8)-32008(10)-14549(12) 
FIG.  ii&?. 


§157.  Other  Trusses.  —  In    Figs.  119,    120,   and    121,  we 
have  examples  of  the  double-panel  system  with  the  load  placed 


OTHER    TRUSSES. 


215 


at  the  lower  panel  points  only.  When,  as  in  119  and  120,  the 
number  of  panels  is  odd,  the  same  ambiguity  arises  as  took  place 
in  Fig.  118.  When,  on  the  other  hand,  the  number  of  panels 
is  even,  as  shown  in  Fig.  121,  there  is  only  one  mode  of  division 
into  systems  possible.  The  diagrams  speak  for  themselves,  and 
need  no  explanation. 


24         6        8       10       12      14       16       18      20       22       24      26      28       30 


1         8       5         7        9       11       13       15       17      19       21      23       25      27      29      31        33        34 


24  8  12  16  20  24 


13  17 

FIG. 


25  29  33     34 


22  26  30       32 


7  11  15  19  23  27  31       38       34 

•FlG.  no*. 


2        4 


22 26 30       32 


5  9  13  17      19  23  27  31      33       31 

FIG.  ngc. 


2  6  10 


24  28  32 


1  11  15  21  26  2»  3S       14 

Fro.  norf. 


216 


APPLIED  MECHANICS. 


246        8       10      12       14       16     18      20      22      24      26 


2       4 


12  16 


24  28  32 


11  15  19  23  27 

FIG.  izoa. 


35 


6  10  14  18  22  26  30  S4      36 


159 


13  17  21  25 

FIG.  i2o£. 


33       35 


6  10  14 


24  28  32  36 


2       4 


26 30 34       36 


13  7  11     "          15  21  25  29  33      35 

FIG.  izod. 


2        4        6        8       10       12       14     16       18 


1       3        5        7        9       11       13      15     17        19      20 


FINK'S   TRUSS. 


217 


2 6 10 14  18 


11  15 

FIG.  i2i«. 


12 16       18 


V 

/      \ 


5  9  13  17 

FIG.  i2i3. 


The  trusses  given  above  may  be  considered  as  examples,  to 
be  solved  by  the  student  by  assuming  the  dead  and  the  moving 
load  per  panel  point  respectively. 

§158.  Fink's  Truss.  —  The  description   of   this  truss  will 
be  evident  from  the  figure.     There  is,  first,  the  primary  truss 
1-8-16;  then  on  each  side 
of  9-8  (the  middle  post  of 
this    truss)    is  a  secondary 
truss    (1-4-9   on    tne    left, 


and  9-12-16  on  the  right). 

Each  of  these  secondary 
trusses  contains  a  pair  of  smaller  secondary  trusses,  and  the 
division  might  be  continued  if  the  segments  into  which  the 
upper  chord  is  thus  divided  were  too  long. 

Of  the  inclined  ties,  there  is  none  in  which  any  load  tends 
to  produce  compression ;  in  other  words,  every  load  either  in- 
creases the  tension  in  the  tie,  or  else  does  not  affect  it.  Hence 


218 


APPLIED   MECHANICS. 


the  greatest  stresses  in  all  the  members  will  be  attained  when 
the  entire  travelling-load  is  on  the  truss,  and  we  need  only  con- 
sider that  case. 

The  determination  of  the  stress  in  any  one  member  can 
readily  be  obtained  by  determining,  by  means  of  the  triangle 
of  forces,  the  stress  in  that  member  due  to  the  presence  of 
the  total  load  per  panel  point,  at  each  point,  and  then  adding  the 
results.  This  will  be  illustrated  by  a  few  diagonals. 

Let  angle  8-1-9  —  *'> 
Let  angle  4-1-5  =  in 
Let  angle  2-1-3  —  **  \ 

we  shall  have,  if  w  -{-  w^  —  entire  load  per  panel  point,  — 


Designation 
of  Ties. 

EFFECT  OF  LOADS  AT 

Resultant 
Tensions. 

3 

5 

7 

9 

11 

13 

15 

1-2 

2-5 
5-6 
6-9 
1-4 
4-9 
1-8 

W  +  Wi 

O 

o 

0 

o 

w  +  wl 

O 

0 

W  +  Wi 

o 

0 
0 

o 

0 

o 

w  +  Wi 

o 

0 

o 

0 
0 

o 
3  w-fw/t 

0 

o 

0 

o 

0 

o 

W  +  Wi 

0 

O 
0 
0 
0 

o 

W  +  Wi 

8  sin  i 

W  +  Wi 

2  sin  i2 
w  +  wl 

2  sin  z'2 

W  +  Wj 

2  sin  z'2 

0 

o 

w  -f  -Wi 

2  sin  z'2 
w  +  wt 

2  sin  z'2 

W  +  Wi 

2  sin  /'2 
w  +  wx 

2  sin  /2 

W  +  Wi 

2  sin  /'2 

W  +   Wi 

4  sin  *x 

W  -\~  Wi 

2  sin  *! 

7f  -}-  Wx 

4  sin  z'i 

W  +  Wi 

sin  /! 
ze/  +  Wi 

4  sin  /i 

W  +  Wi 

2  sin  z'i 

W  +  Wi 

4  sin  /'x 

SW+Wi 

sin  /j      , 

2(«/+W,) 

8  sin  * 

4  sin  / 

8    sin  z 

2  sin  2 

8    sin* 

4  sin  / 

sin  2      i 

i 

The  stresses  in  all  the  other  members  may  be  found  in  a 
similar  manner. 


GENERAL   REMARKS. 


2I9 


§  159.  Bollman's  Truss.  —  The  description  of  this  truss  is 
made  sufficiently  clear  by  the  figure.  The  upper  chord  is  made 
in  separate  pieces  ;  and 

1  3  5  7  9  11  12 

the  short  diagonals  2-5, 
3-4,  4-7,  5-6,  7-8,  6-9, 
8-1 1,  and  9-10  are  only 
needed  to  prevent  a 
bending  of  the  upper 
chord  at  the  joints.  FlG- I24> 

When  this  is  their  only  object,  the  stress  upon  them  cannot  be 
calculated  :  indeed,  it  is  zero  until  bending  takes  place ;  and 
then  it  is  the  less,  the  less  the  bending.  Hence,  in  this  case, 
the  stress  is  wholly  taken  up  by  the  principal  ties  ;  and  these 
have  their  greatest  stress  when  the  whole  load  is  on  the  bridge. 
The  computation  of  the  stresses  is  made  in  a  similar  man- 
ner to  that  used  in  the  Fink. 


§  1 60.  General  Remarks.  —  The  methods  already  explained 
are  intended  to  enable  the  student  to  solve  any  case  of  a  bridge- 
truss  where  there  is  no  ambiguity  as  to  the  course  pursued  by 
the  stresses. 

In  cases  where  a  large  number  of  trusses  of  one  given  type 
are  to  be  computed,  it  would,  as  a  rule,  be  a  saving  of  labor  to 
determine  formulae  for  the  stresses  in  the  members,  and  then 
substitute  in  these  formulae. 

Such  formulae  may  be  deduced  by  using  letters  to  denote 
the  load  and  dimensions,  instead  of  inserting  directly  their 
numerical  values ;  and  then,  having  deduced  the  formulae  for 
the  type  of  truss,  we  can  apply  it  to  any  case  by  merely  sub- 
stituting for  the  letters  their  numerical  values  corresponding 
to  that  case. 

Such  sets  of  formulae  would  apply  merely  to  specific  styles 
of  trusses,  and  any  variation  in  these  styles  would  require  the 
formulae  to  be  changed. 


220  APPLIED  MECHANICS. 

In  order  to  show  how  such  formulas  are  deduced,  a  few  will 
be  deduced  for  such  a  bridge  as  is  shown  in  Fig.  1 1 1. 

Let  the  load  be  applied  at  the  upper  panel  points  only ;  let 
dead  load  per  panel  point  =  w,  moving  load  per  panel  point 
=  w,.  Let  the  whole  number  of  panels  be  N,  N  being  an  even 
number.  Let  the  length  of  one  panel  =  height  of  truss  =  /. 
Then  length  of  entire  span  =  Nl. 

Consider  the  (n  +  i)th  panel  from  the  middle. 

The  stress  in  the  main  tie  is  greatest  when  the  moving  load 
is  on  all  the  panel  points  from  the  farther  abutment  up  to  the 
panel  in  question,  (n  +  0th- 

Hence,  for  the  ntb  panel  from  the  middle,  the  greatest  shear' 
ing-force  that  causes  tension  in  the  main  tie  is  equal  to 

w-\-w1 


Hence  stress  in  main  tie 

N 


For  the  counterbrace,  we  should  obtain,  in  a  similar  way,  the 
formula 


_N 

H   T        \        ^~    1     I  \  fl     I 

2N  LA  2  /  2 


«]  -  wN(»n  +  ,) } , 


which  represents  tension  when  it  is  positive.     Proceed   in  a 
similar  way  for  the  other  members. 

When  there  is  more  than  one  system,  we  must  divide  the 
truss  into  its  component  systems;  and  when  there  is  ambiguity, 
we  must  use,  in  determining  the  dimensions  of  each  member, 
the  greatest  stress  that  can  possibly  come  upon  it. 


CENTRE   OF  GRAVITY.  221 


CHAPTER  V. 

CENTRE    OF   GRAVITY. 

§  161.  The  centre  of  gravity  of  a  body  or  system  of  bodies,  is 
that  point  through  which  the  resultant  of  the  system  of  parallel 
forces  that  constitutes  the  weight  of  the  body  or  system  of 
bodies  always  passes,  whatever  be  the  position  in  which  the 
body  is  placed  with  reference  to  the  direction  of  the  forces. 

§  162.  Centre  of  Gravity  of  a  System  of  Bodies.  —  If 
we  have  a  system  of  bodies  whose  weights  are  Wiy  W2y  Wy  etc., 
the  co-ordinates  of  their  individual  centres  of  gravity  being 
fo,  y»  *i),  (*»  y»  **}>  (*y  Iv  *3)»  etc.,  respectively,  and  if  we 
denote  by  xm  y0,  z0,  the  co-ordinates  of  the  centre  of  gravity  of 
the  system,  we  should  obtain,  just  as  in  the  determination  of  the 
centre  of  any  system  of  parallel  forces,  — 

i°.  By  turning  all  the  forces  parallel  to  OZ>  and  taking 
moments  about  OY, 


(W,  +  W2  +  W3  +  etc.)*0  =   W,x,  +  W2x2  +  Wtx3  +  etc., 
or 


and,  taking  moments  about  OX, 

etc., 


or 


222 


APPLIED   MECHANICS. 


2°.  By  turning  all   the  forces  parallel   to  OX,  and  taking 
moments  about  OY, 


(W*  +  W,  +  Wz  +  etc.K  =  W&  +  W2z2  +  Wzzz  +  etc., 


or 


Hence  we  have,  for  the  co-ordinates  of  the  centre  of  gravity 
of  the  system, 


EXAMPLES. 

I.   Suppose  a  rectangular,  homogeneous  plate  of  brass  (Fig.  125), 

where  AD  =  1 2  inches,  AB  =  5  inches, 
and  whose  weight  is  2  Ibs.,  to  have 
weights  attached  at  the  points  A,  B,  C, 
and  D  respectively,  equal  to  8,  6,  5,  and 
—x  $  Ibs. ;  find  the  centre  of  gravity  of  the 
system. 


4- 


Solution. 

Assume  the  origin  of  co-ordinates  at 
the  centre  of  the  rectangle,  and  we  have 

W,  =  2,  W2  =  8,  W,  =  6,  W4  =  5,  Ws  =  3, 
*,  =o,  x2  =  6,  xz  =  6,  *4  =  -6,  ^s  =  -6, 
Ji  =o,  ^2  =  f,  J3  =  -f,  y4  =  -|,  js  =  f ; 

=  o  -f  48  -f  36  —  30.0  —  18.0  =  36, 
=  o  -f-  20  —  15  —  12.5  4-  7.5  =  o, 
=  2  -f-  8  -f  6  +  5.0  4-  3.0  =  24; 

_  3^>  _  ^ 

=    24=  =24   = 

Hence  the  centre  of  gravity  is  situated  at  a  point  E  on  the  line  OX, 
where  OE  =  1.5. 


CENTRE   OF  GRAVITY  OF .  HOMOGENEOUS  BODIES.     22$ 

2.  Given  a  uniform  circular  plate  of  radius  8,  and  weight  3  Ibs. 
(Fig.  126).  At  the  points  A,  B,  C,  and  D, 
weights  are  attached  equal  to  10,  15,  25,  and  23 
Ibs.  respectively,  also  given  AB  =  45°,  BC  = 
105'',  CD  =  120° ;  find  the  centre  of  gravity  of 
the  system. 


§  163.  Centre  of  Gravity  of  Homogeneous  Bodies.  —  For 
the  case  of  a  single  homogeneous  body,  the  formulae  have  been 
already  deduced  in  §  44.  They  are 


fxdV 

~  JdV 


and  for  the  weight  of  the  body, 

W  =  wfdV, 

where  x&  y0,  zm  are  the  co-ordinates  of  the  centre  of  gravity  of 
the  body,  W  its  weight,  and  w  its  weight  per  unit  of  volume. 

From  these  formulae  we  can  readily  deduce  those  for  any 
special  cases  ;  thus,  — 

(a)  For  a  volume  referred  to  rectangular  co-ordinate  axes, 
d  V  —  dxdydz. 

x    _=  fffxdxdydz  _=  fffydxdydz  =  fffzdxdydz 

SSfdxdydz'    y°    "   Sffdxdydz*     *°  ==  fffdxdydz 


(b)  For  a  flat  plate  of  uniform  thickness,  t,  the  centre  of  grav- 
ity is  in  the  middle  layer;  hence  only  two  co-ordinates  are 
required  to  determine  it.  If  it  be  referred  to  a  system  of  rect- 
angular axes  in  the  middle  plane,  dV  '=  tdxdy, 

_  ffxdxdy  _  ffydxdy 


224  APPLIED  MECHANICS. 

The  centre  of  gravity  of  such  a  thin  plate  is  also  called  the 
centre  of  gravity  of  the  plane  area  that  constitutes  the  middle 
plane  section  ;  hence  — 

(c)  For  a  plane  area  referred  to  rectangular  co-ordinate  axes 
in  its  own  plane, 

Sfxdxdy  ffydxdy 


(d)  For  a  slender  rod  of  uniform  sectional  area,  a,  if  x,  y,  z, 
be  the  co-ordinates  of  points  on  the  axis  (straight  or  curved)  of 
the  rod,  we  shall  have  dV  —  ads  —  a^(dx)2  +  (dy)*  +  (dzf> 


(I)* 


fyds 

=    l-± = 

fds 


A* 


(e)  For  a  slender  rod  whose  axis  lies  wholly  in  one  plane, 
the  centre  of  gravity  lies,  of  course,  in  the  same  plane  ;  and  if 
our  co-ordinate  axes  be  taken  in  this  plane,  we  shall  have  z  =  c 

-=-  =r  o,  and  also  ZQ  =  o.     Hence  we  need  only  two  co 
ax 


CENTRE   OF  GRAVITY  OF  HOMOGENEOUS  BODIES.      22$ 

ordinates  to  determine  the  centre  of  gravity,  hence  dV  •=.  ads 
m  fxds  _  J 


AMI) 


+  }<tx 


* 


4  \dx 


(/)  For  a  line,  straight  or  curved,  which  lies  entirely  in  one 
plane,  we  shall  have,  again, 


+ 


Sds 


fds 

'  ^  }  doc 


Whenever  the  body  of  which  we  wish  to  determine  the 
centre  of  gravity  is  made  up  of  simple  figures,  of  which  we 
already  know  the  positions  of  the  centres  of  gravity,  the  method 
explained  in  §  162  should  be  used,  and  not  the  formulae  that 
involve  integration  ;  i.e.,  taking  moments  about  any  given  line 
will  give  us  the  perpendicular  distance  of  the  centre  of  gravity 
from  that  line. 

In  the  case  of  the  determination  of  the  strength  and  stiff- 
ness of  beams,  it  is  necessary  to  know  the  distance  of  a  hori- 
zontal line  passing  through  the  centre  of  gravity  of  the  section, 


226  APPLIED   MECHANICS. 

from  the  top  or  the  bottom  of  the  section  ;  but  it  is  of  no  prac- 
tical importance  to  know  the  position  of  the  centre  of  gravity 
on  this  line.  In  most  of  the  examples  that  follow,  therefore, 
the  results  given  are  these  distances.  These  examples  should 
be  worked  out  by  the  student. 

In  the  case  of  wrought-iron  beams  of  various  sections,  on 
account  of  the  thinness  of  the  iron,  a  sufficiently  close  approxi- 
mation is  often  obtained  by  considering  the  cross-section  as 
composed  of  its  central  lines ;  the  area  of  any  given  portion 
being  found  by  multiplying  the  thickness  of  the  iron  by  the 
corresponding  length  of  line,  the  several  areas  being  assumed 
to  be  concentrated  in  single  lines. 

EXAMPLES. 

i.  Straight  Line  AB  (Fig*  127).  —  The  centre  of  gravity  is  evidently 
at  the  middle  of  the  line,  as  this  is  a  point  of 


FIG.  127. 


-B    symmetry. 


2.   Combination  of  Two  Straight  Lines.  —  The  centre  of  gravity  in 
each  case  lies  on  the  line  OOI}  Figs.  128,  129,  130,  and  131. 

(a)  Angle- Iron  of  Unequal  Arms  (Fig.  128).  —  Length  AB  =  b, 
length  BC  =  h,  area  AB  •=  A,  area 

BC  =  B; 

q _^^     E;\ p, 

/.    BE  =  DE  =  %  .  A  D"C" 

\Jb2  -f-  h2  FIG.  128. 

(£)  Angle-Iron  of  Equal  Arms  (Fig.  129).  —  Length  AB  =  BC 
B  =  b; 

\^ 

b         b 


FIG.  129. 


CENTRE   OF  GRAVITY  OF  HOMOGENEOUS  BODIES.      22 / 


(c)    Cross  of  Equal  Arms  (Fig.  130).  —  AB  «=  OOt  =  h; 


:.    AC  =  BC 


c p, 


B 

FIG  130. 

(d)  T-Iron  (Fig.  131).  —  Area  AB  =  A,  area  CE  =  B,  length 
A        E        B  CE  =  h; 

Bh 


Fi?.  131.  2  ^    -}-  &) 

3.   Combination  of  Three  Lines.—  OOl  =  line  passing  through  the 
centre  of  gravity. 

(a)   Thin   Isosceles   Triangular  Cell  (Fig.  132).  —  Length 
BC  =  a,  length  AC  =  b,  area  AB  =  BC           A         D          c 
=  ^  area  ^4C  =  B;  o XZ        "7 


/.    DB 


B 

FIG.  132. 


-(-  B 


BE   - 


Same  in  Different  Position  (Fig.  133). 


BD  =  DC  =  - 


FIG.  i  J3. 


228 


APPLIED   MECHANICS. 


(c)   Channel-Iron  (Fig.  134).  —  Area  of  flanges  =  A,  area  of  web 
=  B,  depth  of  flanges  +  J  thickness  of 


Ah 


FIG.  134. 


(d}   H-Beam    (Fig.  135).  —  Area  of  upper  flange  =  AI9  area  of 
lower  flange  =  A2,  area  of  web  =  B,  height  =  h. 

42  +  B 


_ 
= 


k 


C    B 


EOF 

FIG.  135. 


2  As  +  A2  +  B 

4.   Combination  of  Four  Lines. —  OO^  =  line  passing  through  the 
centre  of  gravity. 

(a)    Thin  Rectangular  Cell  (Fig.  136).  — Length  AB  =  h; 


/.     AE  =  BE  =  - 

2 


FIG.  136. 

(£)    Thin  Square  Cell  (Fig.  137). 


Z?  77  /~*  Z7 

•  •          yj  y^     ^^      OA>     ——     — * 

2 


=  BC  = 


-O, 


FIG.  137. 

5.   Circular  Arcs. 

(a)    Circular  Arc  AB  (Fig.  138).  —  Angle  A  OB  =  0,,  radius  =  r, 


Use  formula 


fyds 


FIG.  i3i. 


"   fds  '  -   fds  ' 

but  use  polar  co-ordinates,  where 

ds  =  rdQ,     sf  =•  rcosO,    y  =  rsinO, 


CENTRE   OF  GRAVITY  OF  HOMOGENEOUS  BODIES.     229 


r2       cos  OdO 


2  f 


•i 


Oi 

si 


sin  OdB 


(i  —  cos0,) 


Circular  Arc  AC  (same  figure). 
r  sin  #! 


,    y<>  =  o. 


(c)   Quarter-Arc  of  Circle  AB,  Radius  r  (Fig.  139). 
r2  I  2cos  OdB 

Jo 


2r 


Semi-circumference  ABC  (same  figure). 


FIG-  13* 


6.   Combination  of  Circles  and  Straight  Lines. 

Barlow  Rail  (Fig.    140). — Two  quadrants,  radius   r,  and  web, 
c,        whose  area  =  ^-  the  united  area  of  the  quadrants. 
Let  united  area  of  quadrants  =  A,  area  of  web 
;  let 


AI — IB 

FIG.  140. 


230 


APPLIED   MECHANICS. 


7.  Areas. 

(a)   T-Section  (Fig.  141).  —  Let  length  AB  =  B,  EF  =  b,  entire 
B  height  =  H,  GE  =  h.     Let  distance  of  centre 

x.*mmmmmm^i        of  gravity  below  AB  =  xt;   therefore,  taking 
moments  about  AB  as  an  axis, 
-h(B-t)\ 

•-$ 


-  k(S  - 


whence  we  can  readily  derive  xt. 


(b)  I-Section  (Fig.  142).  — Let  AB  =  B,  GH  =  b,  MN  =  bn 
entire  height  =  H,  BC  =  H  —  h,  EH  =  ht ;  and  let  xt  =  distance 
of  centre  of  gravity  below  AB.  A 

Hence,  taking  moments  about  AB,  we  have 


Xl\B(H  -  h) 
B 


-  h,) 


whence  we  can  deduce  xt 


(c}  Triangle  (Fig.  143).  —  If  we  consider  the  triangle  OBC  as 
composed  of  an  indefinite  number  of  narrow  strips  parallel  to  the  side 
CB,  of  which  FLHK  is  one,  the  centre  of  gravity 
of  each  one  of  these  strips  will  be  on  the  line  OD 
drawn  from  O  to  the  middle  point  of  the  side 
CB  ;  hence  the  centre  ^f  gravity  of  the  entire  tri- 
c  angle  must  be  on  the  line  OD.  For  a  similar  rea- 
son, it  must  be  on  the  median  line  CE  ;  hence  the 
centre  of  gravity  mhst  be  at  the  intersection  of  the  median  lines,  and 
hence 

BC  .  ODsv^ODC 


o 

FIG.  143. 


XQ  = 


=  ^OD.     Moreover,  area  = 


CENTRE   OF  GRAVITY  OF  HOMOGENEOUS  BODIES. 


(d)    Trapezoid  (Fig.  144). 


First  Solution.  —  Bisect  AB  in  6>,  and  CE  in  £>;  let  g^  be  the 


centre  of  gravity  of  CEJB,  and  gt  that  of  ABC. 
Then  will  61,  the  centre  of  gravity  of  the  trape- 
zoid,  be  on  the  line  g^^  and 


Gg,  -. 


Gg,      CEB* 


But  it  must  be  on  the  line  OD;  hence  it  is  at  their  intersectioa 
From  the  similarity  of  GGlgl  GG^g^  we  have 


GG, 

GG  ~ 


ABC 


BEC  ~  CE 


B  m 
b  ; 


GG, 


andsince 


OD 


Second  Solution. — Fig.  144  (a).  Let  O  be  the 
point  of  intersection  of  the  non-parallel  sides  AC 
and  BE.  Let  OF  =  xlt  OD  —  *„  OG  =  xn.  Take 
moments  about  an  axis  through  O,  and  perpen- 
dicular to  OF)  and  we  readily  obtain 


Fie. 


232  APPLIED   MECHANICS. 

(e)  Parabolic   Half-Segment   OAB    (Fig.    145).— Let    OA  ~  x,, 

AB  =  jh  ;  let  x0,y0,  be  the  co-ordinates  of  the  centre  of  gravity  ;  let 

the  equation  of  the  parabola  be  y2  = 

Y 


'.r,      /-'2*M  f»Xl    3 

Jo    xdocdy     2a  J0  **d 


X0  = 


/*'  r°ya 

t/o        t/o         ' 


r<£^     /».r,  ^       3    ^ 

t/o     y  ~~ s    i  ~  3v     i  MI  — 


Area 


(/)  Parabolic  Spandril  OBC  (Fig.  145).  —  Let  x0,y0,  be  co-ordi- 
nates of  centre  of  gravity  of  the  spandril. 


xdxdy 


x^   (*y\ 
I 

_  l_?«y 
/*J«r,      /*yt 

/  /  L 

1/0        ^ 


_ 

>       /»*,  /»^ 

/  / 

1/0    ^  2* 
Area  =  ^j,  - 


CENTRE  OF  GRA  VI TY  OF  HOMOGENEOUS  BODIES.       233 


(g)  Circular  Sector  OAC  (Fig.  146).— Let  OA  =  r,  AOX  =  0,, 
be  the  co-ordinates  of  the  centre  of  gravity  : 


.  .       ^o^O, 


Xo   = 


/r  r*V  r"*  —  x*  /»rcos0j      /*jr  tan  0, 

/  *<**#+    /  /  JCflkfljK 

.ooggy-V^rra  y°  ^-rtanfl,      ^ 


Area  — 


c 

FIG.  146. 


Second  Solution. 

Consider  the  sector  to  be  made  up  of  an  indefinite  number  of 
narrow  rings  ;  let  p  be  the  variable  radius,  and  dp  the  thickness  : 

Elementary  area  =  2pftldp, 

and  centre  of  gravity  of  this  elementary  area  is  on  OX,  at  a  distance 
from  O  equal  to  p — ^ — 1  [see  Example  5  (£)]  ; 


X  = 


(ti)   Circular  Half-Segment  ABX  (Fig.  146), 


f"      fQ   "  xdxdy  f" xVr*  -  x*dx 

Sector  minus  triangle       %r*Sl  —  |r2  sin  0,  cos  0, 


,  —  sn  <,  cos 


r    r^ 

^rcosgyo       '"""'      =  .  4sinBig1-sina0.cosg1 
,  —  sin  0,  cos  0,)  ~~  0,  —  sin  0,  cos  0, 


234  APPLIED  MECHANICS. 

§  164.  Pappus's  Theorems.  —  The  following  two  theorems 
serve  often  to  simplify  the  determination  of  the  centres  of 
gravity  of  lines  and  areas.  They  are  as  follows  :  — 

THEOREM  I.  —  If  a  plane  curve  lies  wholly  on  one  side  of  a 
straight  line  in  its  own  plane,  and,  revolving  about  that  line, 
generates  thereby  a  surface  of  revolution,  the  area  of  the  sur- 
face is  equal  to  the  product  of  the  length  of  the  revolving  line, 
and  of  the  path  described  by  its  centre  of  gravity. 

Proof.  —  Let  the  curve  lie  in  the  xy  plane,  and  let  the  axis 
of  y  be  the  line  about  which  it  revolves.  We  have,  from  what 

fxds 
precedes,  §  163  (e\  XQ  =-         -' 


.*.     x0fds  =  fxds, 

where  x0  equals  the  perpendicular  distance  of   the  centre  of 
gravity  of  the  curve  from  O  Y,  ds  =  elementary  arc, 

2irx0fds  =  f(2irx)ds; 
or,  reversing  the  equation, 

f(2irx)ds  = 


But  f(2irx)ds  =  surface  described  in  one  revolution,  while  s  =. 
length  of  arc,  and  2irxQ  =  path  described  by  the  centre  of 
gravity  in  one  revolution.  Hence  follows  the  proposition. 

THEOREM  II.  —  If  a  plane  area  lying  wholly  on  the  same 
side  of  a  straight  line  in  its  own  plane  revolves  about  that  line, 
and  thereby  generates  a  solid  of  revolution,  the  volume  of  the 
solid  thus  generated  is  equal  to  the  product  of  the  revolving 
area,  and  of  the  path  described  by  the  centre  of  gravity  of  the 
plane  area  during  the  revolution, 


PAPPUS'S   THEOREMS.  235 

Proof.  —  Let  the  area  lie  in  the  xy  plane,  and  let  the  axis 
OY  be  the  axis  of  revolution.  We  then  have,  from  what  has 
preceded,  if  x0  =  perpendicular  distance  of  the  centre  of  gravity 
of  the  plane  area  from  OYt  the  equation,  §  163  (b), 

Sfxdxdy 
*°-  ffdxdy' 
Hence 

Xo  ffdxdy          =  ffxdxdy; 

/.     (2irx0)  ffdxdy 
or 

ff(2trx)dxdy 


But  ff(2irx)dxdy  =  volume  described  in  one  revolution,  and 
2iexQ  =  path  described  by  the  centre  of  gravity  in  one  revolu- 
tion. Hence  follows  the  proposition. 

The  same  propositions  hold  true  for  any  part  of  a  revolution, 
as  well  as  for  an  entire  revolution,  since  we  might  have  multi- 
plied through  by  the  circular  measure  6,  instead  of  by  2ir. 

It  is  evident  that  the  first  of  these  two  theorems  may  be 
used  to  determine  the  centre  of  gravity  of  a  line,  when  the 
length  of  the  line,  and  the  surface  described  by  revolving  it 
about  the  axis,  are  known  ;  and  so  also  that  the  second  theorem 
may  be  used  to  determine  the  centre  of  gravity  of  a  plane  area 
whenever  the  area  is  known,  and  also  the  volume  described  by 
revolving  it  around  the  axis. 

EXAMPLES. 

i.  Circular  Arc  AC  (Fig.  138).  —  Length  of  arc  =  s  =  2rO,  sur- 
face of  zone  described  by  revolving  it  about  O  Y  =  circumference  of  a 
great  circle  multiplied  by  the  altitude  =  (ztrr)  (2rsmOl); 


x00l  =  rsinfl, 


sm0z 
r- 


236  APPLIED   MECHANICS. 

2.  Semicircular  Arc  (Fig.  139).  —  Length  of  arc  =  nr,  surface  of 
sphere  described  =  4?rr2 ; 

2r 
.'.     2Trx0(Trr)  =  4?rr2  .*.     x0  =  — • 

7T 

3.  Trapezoid  (Fig.  147).  —  Let  AD  =  b,  BC  —  b ;  let  it  revolve 
around  AD :  it  generates  two  cones  and  a  cylinder. 

AD  +  BC 
Y  Area  of  trapezoid  =  -  —BG, 

B   Volume  =  ~ -(AG  +  HD)  +  7r(G£)2 .  BC 


\-HD+  3BC) 
FlG-  J47.  =  ^ ^(^Z)  +  BC  +  ^C) 


GBI  BC       \       GBI 

=  KL 


4.    Circular  Sector  AGO    (Fig.  146).  —  Area   of  sector  =   r*0lt 
volume    described  =   -Jr  (surface    of   zone)    =   \r(2-rrr}  (?r  sin  0,)    = 
sin  0! 


§  165.  Centre  of  Gravity  of  Solid  Bodies.  —  The  general 
formulae  furnish,  in  most  cases,  a  very  complicated  solution,  and 
hence  we  generally  have  recourse  to  some  simpler  method.  A 
few  examples  will  be  given  in  this  and  the  next  section. 


CENTRE   OF  GRAVITY  OF  SYMMETRICAL   BODIES.      237 


Tetrahedron  ABCD  (Fig.  148). — The  plane  ABE,  containing  the 
edge  AB  and  the  middle  point  E  of  the  edge  CD,  bisects  all  lines 
drawn  parallel  to  CD,  and  terminating  in  the  faces  A 

ABD  and  ABC :  hence  a  similar  reasoning  to  that 
used  in  the  case  of  the  triangle  will  show  that  the  cen- 
tre of  gravity  of  the  pyramid  must  be  in  the  plane 
ABE ;  in  the  same  way  it  may  be  shown  that  it  must 
lie  in  the  plane  ACF.  Hence  it  must  lie  in  their 
intersection,  or  in  the  line  AG  joining  the  vertex  A 
with  the  centre  of  gravity  (intersection  of  the  medians) 
of  the  opposite  face. 


FIG.  148. 
In  the  same  way  it  can  be  shown  that  the  centre 


of  gravity  of  the  triangular  pyramid  must  lie  in  the  line  drawn  from 
the  vertex  B  to  the  centre  of  gravity  of  the  face  A  CD.  Hence  the 
centre,  of  gravity  of  the  tetrahedron  will  be  found  on  the  line  AG  at 
a  distance  from  G  equal  to  \A  G. 

§  1 66.  Centre  of  Gravity  of  Bodies  which  are  Symmet- 
rical with  Respect  to  an  Axis.  —  Such  solids  may  be  gener- 
ated by  the  motion  of  a  plane  figure,  as  ABCD 
(Fig.  149),  of  variable  dimensions,  and  of  any 
form  whose  centre  G  remains  upon  the  axis 
OX ;  its  plane  being  always  perpendicular  to 
OX,  and  its  variable  area  X  being  a  function 
of  x,  its  distance  from  the  origin. 

Here  the  centre  of  gravity  will  evidently 
FIG.  i49.  jje  on  tne  axjs  QX^  and  the  elementary  vol- 

ume will  be  the  volume  of  a  thin  plate  whose  area  is  X  and 
thickness  A;r  ;  hence  the  elementary  volume  will  be 

Take  moments  about  OY,  and  we  shall  have 


or 


x0fXdx  =  fXxdx     and     Volume  =  fXdx, 
fXxdx 

**=7xJ*"         F=/^» 


APPLIED  MECHANICS. 


EXAMPLES.  • 

x2       y2       z2 

I.  Ellipsoid  — •  -f  ^-  +  —  =  i  (Fig.  150).  —  Find  centre  of  gravity 
a        D        c 

of  the  half  to  the  right  of  the  x  plane.    Let  OK 
=  x.     Now  if,  in  the  equation  of  the  ellipsoid, 

v  X2  Z2 

we  make  y  =  o,  we  have  —  H =  I ; 


where  z  = 

Make  z  =  o  in  the  equation  of  the  ellipsoid,  and  —  +  ij  =  I  > 


where  ^  = 


.-.    EK 


are  the  semi-axes  of  the  variable  ellipse  EGFH,  which,  by  moving  along 
OX,  generates  the  ellipsoid.     Hence 


hence 


irbc 
Area  EGFH  =  Tr(EK .  GK)  =  —(a*  -  x2)  =  X; 


Elementary  volume  =  —  (a2  — 


irbc  (*a  .  ( a2x2       x4 ) 

—  I     (a2x  —  x*)dx  < > 

a2  Jo     _  (    2  4  ). 

trbc 


^  J  (  x*}a 

a2-x*)dx  \a2x--\ 

3)0 

V   =  -  -  I  a  (a2  -  x2)dx     =  \irabc. 

d2  t/0 

a.  Hemisphere.  —  Make  a  =  b  =  c,  and  x0  =  f a,  V 


CENTRE   OF  GRAVITY  OF  SYMMETRICAL  BODIES.      239 

If  the  section  X  were  oblique  to  OXt  making  an  angle  0. 
with  it,  the  elementary  volume  would  not  be  Xdx,  but  Xdx  sin  0, 

and  we  should  have 


3.  Oblique  Cone  (Fig.  151).  —  Let  OA  =  h;  let  area  of  base  be 
and  let  the  angle  made  by  OX  with  the  base  be  6; 

X      x>  A 


FIG.  151. 


rh 

sin0  /    ^^ 

**  o 


4.  Truncated  Cone   (Fig.  151).  —  Let   height   of  entire  cone  be 
h  =  OA  ;  let  height  of  portion  cut  off  be  hl ; 

AT*  h*-  h* 

—  I     x*dx       

4  ,/^-A4 


^TTSi 


240  APPLIED   MECHANICS. 


CHAPTER  VI. 

STRENGTH  OF  MATERIALS. 

§  167.  Stress,  Strain,  and  Modulus  of  Elasticity When 

a  body  is  subjected  to  the  action  of  external  forces,  if  we 
imagine  a  plane  section  dividing  the  body  into  two  parts,  the 
force  with  which  one  part  of  the  body  acts  upon  the  other 
at  this  plane  is  called  the  stress  on  the  plane  ;  it  may  be  a 
tensile,  a  compressive,  or  a  shearing  stress,  or  it  may  be  a  com- 
bination of  either  of  the  two  first  with  the  last.  In  order  to 
know  the  stress  completely,  we  must  know  its  distribution  and 
its  direction  at  each  point  of  the  plane.  If  we  consider  a  small 
area  lying  in  this  plane,  including  the  point  O,  and  represent 
the  stress  on  this  area  by  /,  whereas  the  area  itself  is  repre- 
sented by  a,  then  will  the  limit  of  <-  as  a  approaches  zero  be  the 

a 

intensity  of  the  stress  on  the  plane  under  consideration  at  the 
point  0. 

When  a  body  is  subjected  to  the  action  of  external  forces, 
and,  in  consequence  of  this,  undergoes  a  change  of  form,  it 
will  be  found  that  lines  drawn  within  the  body  are  changed,  by 
the  action  of  these  external  forces,  in  length,  in  direction,  or 
in  both ;  and  the  entire  change  of  form  of  the  body  may  be 
correctly  described  by  describing  a  sufficient  number  of  these 
changes. 

If  we  join  two  points,  A  and  B,  of  a  body  before  the 
external  forces  are  applied,  and  find,  that,  after  the  application 
of  the  external  forces,  the  line  joining  the  same  two  points  of 
the  body  has  undergone  a  change  of  length  &(AB),  then  is  the 


STRESS,   STRAIN,   AND   MODULUS   OF  ELASTICITY.      241 

limit  of   the  ratio  '  as  AB  approaches   zero   called   the 


strain  of  the  body  at  the  point  A  in  the  direction  AB. 

If  AB  4-  &(AB)  >  AB,  the  strain  is  one  of  tension. 

If  AB  +  A  (^4-5)  <  ^4-#,  the  strain  is  one  of  compression. 


Suppose  a  straight  rod  of  uniform  section  A  to  be  subjected 
to  a  pull  P  in  the  direction  of  its  length,  and  that  this  pull  is 
uniformly  distributed  over  the  cross-section  :  then  will  the  in- 
tensity of  the  stress  on  the  cross-section  be 


If  P  be  measured  in  pounds,  and  A  in  square  inches,  then  will 
/  be  measured  in  pounds  per  square  inch. 

If  the  length  of  the  rod  before  the  load  is  applied  be  /, 
and  its  length  after  the  load  is  applied  be  I  ~\-  e,  then  is  e  the 
elongation  of  the  rod  ;  and  if  this  elongation  is  uniform  through- 

x> 

out  the  length  of  the  rod,  then  is  -  the  elongation  of  the  rod 

per  unit  of  length,  or  the  strain. 

Hence,  if  a  represent  the  strain  due  to  the  stress  /  per 
unit  of  area,  we  shall  have 


The  Modulus  of  Elasticity  is  commonly  defined  as  the  ratio 
of  the  stress  per  unit  of  area  to  the  strain,  or 

*-*-; 

a 

and  this  is  expressed  in  units  of  weight  per  unit  of  area,  as  in 
pounds  per  square  inch. 

This   definition  is  true,  however,  only  for  stresses  for  which 
Hooke's  law  "  The  stress  is  proportional  to  the  strain  "  holds. 


242  APPLIED    MECHANICS. 

For  greater  stresses  the  permanent  set  must  first  be  deducted 
from  the  strain,  and  the  remainder  be  used  as  divisor. 

The  limit  of  elasticity  of  any  material  is  the  stress  above 
which  the  stresses  are  no  longer  proportional  to  the  strains. 

The  modulus  of  elasticity  was  formerly  defined  as  the 
weight  that  would  stretch  a  rod  one  square  inch  in  section  to 
double  its  length,  if  Hooke's  law  held  up  to  that  point,  and 
the  rod  did  not  break. 

EXAMPLES. 

1.  A  wrought-iron  rod  10  feet  long  and  i  inch  in  diameter  is  loaded 
in  the  direction  of  its  length  with  8000  Ibs. ;  find  (i)  the  intensity  of 
the  stress,  (2)  the  elongation  of  the  rod ;  assuming  the  modulus  of  the 
iron  to  be  28000000  Ibs.  per  square  inch. 

2.  What  would  be  the  elongation   of  a   similar  rod   of  cast-iron 
under  the  same  load,  assuming  the  modulus  of  elasticity  of  cast-iron  to 
be  1 7000000  Ibs.  per  square  inch  ? 

3.  Given  a  steel  bar,  area  of  section  being  4  square  inches,  the 
length  of  a  certain  portion  under  a  load  of  25000  Ibs.  being  10  feet, 
and  its  length  under  a  load  of  100000  Ibs.  being  10'  o".o75  ;  find  the 
modulus  of  elasticity  of  the  material. 

4.  What  load  will  be  required  to  stretch  the  rod  in  the  first  example 
Y1^  inch  ? 

§  1 68.  Resistance  to  Stretching  and  Tearing.  —  The  most- 
used  criterion  of  safety  against  injury  for  a  loaded  piece  is, 
that  the  greatest  intensity  of  the  stress  to  which  any  part  of  it 
is  subjected  shall  nowhere  exceed  a  certain  fixed  amount,  called 
the  working-strength  of  the  material  ;  this  working-strength 
being  a  certain  fraction  of  the  breaking-strength  determined 
by  practical  considerations. 

The  more  correct  but  less  used  criterion  is,  that  the  great- 
est strain  in  any  part  of  the  structure  shall  nowhere  exceed 
the  working-strain  ;  the  greatest  allowable  amount  of  strain 
being  a  fixed  quantity  determined  by  practical  considerations. 


RESISTANCE    TO  STRETCHING   AND    TEARING.          243 

This  is  equivalent  to  limiting  the  allowable  elongation  or 
compression  to  a  certain  fraction  of  its  length,  or  the  deflection 
of  a  beam  to  a  certain  fraction  of  the  span. 

If  the  stress  on  a  plane  surface  be  uniformly  distributed, 
its  resultant  will  evidently  act  at  the  centre  of  gravity  of  the 
surface,  as  has  been  already  shown  in  §  42  to  be  the  case  with 
any  uniformly  distributed  force. 

If  a  straight  rod  of  uniform  section  and  material  be  sub- 
jected to  a  pull  in  the  direction  of  its  length,  and  if  the  result- 
ant of  the  pull  acts  along  a  line  passing  through  the  centres 
of  gravity  of  the  sections  of  the  rod,  it  is  assumed  in  practice 
that  the  stress  is  uniformly  distributed  throughout  the  rod,  and 
hence  that  for  any  section  we  shall  obtain  the  stress  per  square 
inch  by  dividing  the  total  pull  by  the  number  of  square  inches 
in  the  section. 

If,  on  the  other  hand,  the  resultant  of  the  pull  does  not  act 
through  the  centres  of  gravity  of  the  sections,  the  pull  is  not 
uniformly  distributed ;  and  while 


will  express  the  mean  stress  per  square  inch,  the  actual  inten- 
sity of  the  stress  will  vary  at  different  points  of  the  section, 

p 
being  greater  than  —  at  some  points  and  less  at  others.     How 

A 

to  determine  its  greatest  intensity  in  such  cases  will  be  shown 
later. 

With  good  workmanship  and  well-fitting  joints,  the  first 
case,  or  that  of  a  uniformly  distributed  stress,  can  be  practi- 
cally realized ;  but  with  ill-fitting  joints  or  poor  workmanship, 
or  with  a  material  that  is  not  homogeneous,  the  resultant  of 
the  pull  is  liable  to  be  thrown  to  one  side  of  the  line  passing 
through  the  centres  of  gravity  of  the  sections,  and  thus  there 


244  APPLIED   MECHANICS. 

is  set  up  a  bending-action  in  addition  to  the  direct  tension,  and 
therefore  an  unevenly  distributed  stress. 

It  is  of  the  greatest  importance  in  practice  to  take  cogni- 
zance of  any  such  irregularities,  and  determine  the  greatest 
intensity  of  the  stress  to  which  the  piece  is  subjected  :  though 
it  is  too  often  taken  account  of  merely  by  means  of  a  factor  of 
safety  ;  in  other  words,  by  guess. 

Leaving,  then,  this  latter  case  until  we  have  studied  the 
stresses  due  to  bending,  we  will  confine  ourselves  to  the  case 
of  the  uniformly  distributed  stress. 

If  the  total  pull  on  the  rod  in  the  direction  of  its  length 
be  P,  and  the  area  of  its  cross-section  A,  we  shall  have,  for  the 
intensity  of  the  pull, 

P 


On  the  other  hand,  if  the  working-strength  of  the  material 
per  unit  of  area  be  /,  we  shall  have,  for  the  greatest  admissible 

load  to  be  applied, 

P  =  fA. 

If  /  be  the  working-strength  of  the  material  per  square 
inch,  and  E  the  modulus  of  elasticity,  then  is  the  greatest 
admissible  strain  equal  to 


Thus,  assuming  12000  Ibs.  per  square  inch  as  the  working 
tensile  strength  of  wrought-iron,  and  28000000  Ibs.  per  square 
inch  as  its  modulus  of  elasticity,  its  working-strain  would  be 


1  2000 


28000000       7000 

Hence  the  greatest  safe  elongation   of   the   bar  would   be 
of  its  length.     Hence  a  rod   10  feet  long  could  safely  be 
stretched  ^  of  a  foot  =  0.05 14". 


VALUES   OF  BREAKING  AND    WORKING    STRENGTH.    245 

§  169.  Approximate  Values  of  Breaking  Strength,  and 
of  Modulus  of  Elasticity. — In  a  later  part  of  this  book  the 
attempt  will  be  made  to  give  an  account  of  the  experiments 
that  have  been  made  to  determine  the  strength  and  elas- 
ticity of  the  materials  ordinarily  used  in  construction,  in  such 
a  way  as  to  enable  the  student  to  decide  for  himself,  in  any- 
special  case,  upon  the  proper  values  of  the  constants  that  he 
ought  to  use. 

For  the  present,  however,  the  following  will  be  given  as  a 
rough  approximation  to  some  of  these  quantities,  which  we  may 
make  use  of  in  our  work  until  we  reach  the  above-mentioned 
account. 

(a)  Cast-Iron. 

Breaking  tensile  strength  per  square  inch,  of  common  quali- 
ties, 14000  to  20000  Ibs. ;  of  gun  iron,  30000  to  33000  Ibs. 

Modulus  of  elasticity  for  tension  and  for  compression,  about 
17000000  Ibs.  per  square  inch. 

(b)  Wrought-Iron. 

Breaking  tensile  strength  per  square  inch,  from  40000  to* 
60000  Ibs. 

Modulus  of  elasticity  for  tension  and  for  compression,  about 
28000000. 

(c)  Mild  Steel. 

Breaking  tensile  strength  per  square  inch,  55000  to  70000 
Ibs. 

Modulus  of  elasticity  for  tension  and  for  compression,  from 
28000000  to  30000000  Ibs.  per  square  inch. 

(d)  Wood. 

Breaking  compressive  strength  per  square  inch :  — 

Oak,  green 3000  Ibs. 

Oak,  dry 3000  to  6000  Ibs. 

Yellow  pine,  green 3000  to  4000  Ibs. 

Yellow  pine,  dry 4000  to  7000  Ibs. 


246  APPLIED   MECHANICS. 

Modulus  of  elasticity  for  compression  (average  values) :  — 

Oak 1300000  Ibs.  per  square  inch. 

Yellow  pine 1600000  Ibs.  per  square  inch. 

§  170.  Sudden  Application  of  the  Load.  —  If  a  wrought- 
iron  rod  10  feet  long  and  I  square  inch  in  section  be  loaded 
with  12000  pounds  in  the  direction  of  its  length,  and  if  the 
modulus  of  elasticity  of  the  iron  be  28000000,  it  will  stretch 
0.05 14"  provided  the  load  be  gradually  applied :  thus,  the  rod 
begins  to  stretch  as  soon  as  a  small  load  is  applied ;  and,  as  the 
load  gradually  increases,  the  stretch  increases,  until  it  reaches 
0.05 14". 

If,  on  the  other  hand,  the  load  of  12000  Ibs.  be  suddenly 
applied  (i.e.,  put  on  all  at  once)  without  being  allowed  to  fall 
through  any  height  beforehand,  it  would  cause  a  greater  stretch 
at  first,  the  rod  undergoing  a  series  of  oscillations,  finally 
settling  down  to  an  elongation  of  0.05 14". 

To  ascertain  what  suddenly  applied  load  will  produce  at 
most  the  elongation  0.05 14",  observe,  that,  in  the  case  of  the 
gradually  applied  load,  we  have  a  load  gradually  increasing  from 

o  to  12000  Ibs. 

Its  mean  value  is,  therefore,  ^(12000)  =  6000  Ibs.  ;  and  this 
force  descends  through  a  distance  of 

0.05 14". 

Hence  the  amount  of  mechanical  work  done  on  the  rod  by  the 
gradually  applied  load  in  producing  this  elongation  is 

(6000)  (0.0514)  =  308.4  inch-lbs. 

Hence,  if  we  are  to  perform  upon  the  rod  308.4  inch-lbs.  of 
work  with  a  constant  force,  and  if  the  stretch  is  to  be  0.05 14", 
the  magnitude  of  the  force  must  be 

308.4 


0-0514" 


=  6000  Ibs. 


RESILIENCE    OF  A    TENSION-BAR.  247 

Hence  a  suddenly  applied  load  will  produce  double  the  strain 
that  would  be  produced  by  the  same  load  gradually  applied ; 
and,  moreover,  a  suddenly  applied  load  should  be  only  half  as 
great  as  one  gradually  applied  if  it  is  to  produce  the  same 
strain. 

§  171.  Resilience  of  a  Tension-Bar.  —  The  resilience  of  a 
tension-rod  is  the  mechanical  work  done  in  stretching  it  to  the 
same  amount  that  it  would  stretch  under  the  greatest  allowable 
gradually  applied  load,  and  is  found  by  multiplying  the  greatest 
allowable  load  by  half  the  corresponding  elongation. 

Thus,  suppose  a  load  of  100  Ibs.  to  be  dropped  upon  the 
rod  described  above  in  such  a  way  as  to  cause  an  elongation  not 
greater  than  0.05 14",  it  would  be  necessary  to  drop  it  from  a 
height  not  greater  than  3.08". 

EXAMPLES. 

1 .  A  wrought-iron  rod  is  1 2  feet  long  and  i  inch  in  diameter,  and 
is  loaded  in  the  direction  of  its  length;   the  working-strength   of  the 
iron  being  12000  Ibs.  per  square  inch,  and  the  modulus  of  elasticity 
28000000  Ibs.  per  square  inch. 

Find  the  working-strain. 
Find  the  working-load. 
Find  the  working-elongation. 
Find  the  working-resilience. 

From  what  height  can  a  5o-pound  weight  be  dropped  so  as  to  produce 
tension,  without  stretching  it  more  than  the  working- elongation? 

2.  Do  the  same  fora  cast-iron  rod,  where  the  working-strength  is 
5000  pounds  per  square  inch,  and  the  modulus  of  elasticity  17000000; 
the  dimensions  of  the  rod  being  the  same. 

§  172.  Results  of  Wohler's  Experiments  on  Tensile 
Strength.  —  According  to  the  experiments  of  Wohler,  of  which 
an  account  will  be  given  later,  the  breaking-strength  of  a  piece 


248  APPLIED  MECHANICS. 

depends,  not  only  on  whether  the  load  is  gradually  or  suddenly 
applied,  but  also  on  the  extreme  variations  of  load  that  the 
piece  is  called  upon  to  undergo,  and  the  number  of  changes  to 
which  it  is  to  be  submitted  during  its  life. 

For  a  piece  which  is  always  in  tension,  he  determines  the 
following  two  constants  ;  viz.,  /,  the  carrying-strength  per  square 
inch,  or  the  greatest  quiescent  stress .  that  the  piece  will  bear, 
and  u,  the  primitive  safe  strength,  or  the  greatest  stress  per 
square  inch  of  which  the  piece  will  bear  an  indefinite  number 
of  repetitions,  the  stress  being  entirely  removed  in  the  inter- 
vals. 

This  primitive  safe  strength,  uy  is  used  as  the  breaking- 
strength  when  the  stress  varies  from  o  to  u  every  time.  Then, 
by  means  of  Launhardt's  formula,  we  are  able  to  determine  the 
ultimate  strength  per  square  inch  for  any  different  limits  of 
stress,  as  for  a  piece  that  is  to  be  alternately  subjected  to  80000 
and  6000  pounds. 

Thus,  for  Phoenix  Company's  axle  iron,  Wohler  finds 

/  =  3290  kil.  per  sq.  cent.  =  46800  Ibs.  per  sq.  in., 
u  —  2100  kil.  per  sq.  cent.  =  30000  Ibs.  per  sq.  in. 

Launhardt's  formula  for  the  ultimate  strength  per  unit  of  area 
is 

t  —  u     least  stress    ) 


a  =  u{\  + 


u      greatest  stress)' 


Hence,  with  these  values  of  t  and  uy  we  should  have,  for  the 
ultimate  strength  per  square  inch, 

(  i    least  stress    ) 

a  —    2100%  i  -h  -  ->kil.  per  sq.  cent., 

(          2 greatest  stress) 

or 

ii    least  stress    ) 
i  -f-  -  -\ Ibs.  per  sq.  in. 

2 greatest  stress) 


WOH LEWS  EXPERIMENTS   ON   TENSILE  STRENGTH.      249 


Thus,  if  least  stress  =  6000,  and  greatest  =  80000,  we  should 

have 

a  =  30000$ i  -f  £  .  -fo\  =  30000^1  +  isul  =  3II25> 

if  least  stress  =  60000,  and  greatest  =  80000, 

a  =  30000 | i  -f  i  .  f  I  =  30000^1  +  fj  =  41250; 

if  least  stress  =  greatest  stress  =  80000, 

a  —  30000^1  -j-  \\  =  45000  =  carrying-strength. 

Hence,  instead  of  using,  as  breaking-strength  per  square  inch 
in  all  cases,  45000,  we  should  use  a  set  of  values  varying  from 
45000  down  to  30000,  according  to  the  variation  of  stress  which 
the  piece  is  to  undergo. 

For  working-strength,  Weyrauch  divides  this   by  3  :   thus 
obtaining,  for  working-strength  per  square  inch, 

(          i     least  stress    )  „ 

b  —  10000  <  i  H [  Ibs.  per  sq.  in. ; 

(          2  greatest  stress ) 

lor  Krupp's  cast-steel,  notwithstanding  the  fact  that  Wohler 

finds 

/  =  7340  kil.  per  sq.  cent.  =  104400  Ibs.  per  sq.  in., 
u  =  33°°  kil.  per  sq.  cent.  =    46900  Ibs.  per  sq.  in., 

Weyrauch  recommends 

(  q     least  stress    ) 

a  —     3300  \  i  +  -  -Vkil.  per  sq.  cent., 

(          ii  greatest  stress) 

(  o     least  stress    ) 

a  =  46000  <  i  4-  —  —  >lbs.  per  sq.  in., 

(          ii  greatest  stress) 


=  I5633J 


o     least  stress    ) 

i  4-  - ;  -  -  >  Ibs.  per  sq.  in. 

1 1  greatest  stress  j 


EXAMPLES. 

Find  the  breaking-strength  per  square  inch  for  a  wrought-iron  tension 
rod. 

1.  Extreme  loads  are     75000  and      6000  Ibs. 

2.  Extreme  loads  are  120000  and  100000  Ibs. 

3.  Extreme  loads  are  300000  and    10000  Ibs. 
Find  the  safe  section  for  the  rod  in  each  case. 


250  APPLIED  MECHANICS. 

§173.  Suspension-Rod  of  Uniform  Strength.  —  In  the 
case  of  a  long  suspension-rod,  the  weight  of  the  rod  itself  some- 
times becomes  an  important  item.  The  upper  section  must,  of 
course,  be  large  enough  to  bear  the  weight  that  is  hung  from 
the  rod  plus  the  weight  of  the  rod  itself ;  but  it  is  sometimes 
desirable  to  diminish  the  sections  as  they  descend.  This  is  often 
accomplished  in  mines  by  making  the  rod  in  sections,  each  section 
being  calculated  to  bear  the  weight  below  it  plus  its  own  weight. 
Were  the  sections  gradually  diminished,  so  that  each  section 
would  be  just  large  enough  to  support  the  weight  below  it,  we 
should,  of  course,  have  a  curvilinear  form ;  and  the  equation  of 
this  curve  could  be  found  as  follows,  or,  rather,  the  area  of  any 
section  at  a  distance  from  the  bottom  of  the  rod. 
Let  W  =  weight  hung  at  O  (Fig.  152), 

Let  w  =  weight  per  unit  of  volume  of 

the  rod, 

Let  x  =  distance  AO, 

Let  5  =  area  of  section  A, 

Let  x  +  dx  =  distance  BO, 


Let  5  +  dS  =  area  of  section  at  Bt 


Let  f  =  working-strength  of  the  mate- 

rial per  square  inch. 

i°.  The  section  at  O  must  be  just  large  enough 
to  sustain  the  load  W; 
*ff  '  W 

FIG.  152.  •*•      SQ  =  —f- 

2°.  The  area  in  dS  must  be  just  enough  to  sustain   the 
weight  of  the  portion  of  the  rod  between  A  and  B. 
The  weight  of  this  portion  is  wSdx ; 

_       wSdx 
.'.    <£>=    — 

dS       w  iv 

.*.    -=  =  -fdx  /.    log,  S  =  -7X  H-  a  constant 

•>       /  / 


CYLINDERS  SUBJECTED    TO   INTERNAL   PRESSURE.     .2$  I 


W 

When  x  =  o,  5  =  -^  ; 

W  IW\      w 

.-.    log*  Y  =  the  constant  .%    log,.S  —  log,(  -y- )  =  -^ 


This  gives  us  the  means  of  determining  the  area  at  any  dis- 
tance x  from  O. 

EXAMPLES. 

1.  A  wrought-iron  tension-rod  200  feet  long  is  to  sustain  a  load  of 
2000  Ibs.  with  a  factor  of  safety  of  4,  and  is  to  be  made  in  4  sections, 
each  50  feet  long;  find  the  diameter  of  each  section,  the  weight  of  the 
wrought  iron  being  480  Ibs.  per  cubic  foot. 

2.  Find  the  diameter  needed  if  the  rod  were   made   of  uniform 
section,  also  the  weight  of  the  extra  iron  necessary  to  use  in  this  case. 

3.  Find  the  equation  of  the  longitudinal  section  of  the  rod,  assum- 
ing a  square  cross-section,  if  it  were  one  of  uniform  strength,  instead  of 
being  made  in  4  sections. 

§  174.  Thin  Hollow  Cylinders  subjected  to  an  Internal 
Normal  Pressure.  —  Let/  denote  the  uniform  intensity  of  the 
pressure  exerted  by  a  fluid  which  is  confined  within  a  hollow 
cylinder  of  radius  r  and  of  thickness  /  (Fig,  153), 
the  thickness  being  small  compared  with  the  radius. 
Let  us  consider  a  unit  of  length  of  the  cylinder,  and  c( 
let  us  also  consider  the  forces  acting  on  the  upper 
half-ring  CED.  PIG.  153. 

The  total  upward  force  acting  on  this  half-ring,  in  conse- 
quence of  the  internal  normal  pressure,  will  be  the  same  as 
that  acting  on  a  section  of  the  cylinder  made  by  a  plane  pass- 
ing through  its  axis,  and  the  diameter  CD.  The  area  of  this 


252  APPLIED   MECHANICS. 

section  will  be  2r  X  I  =  2r  :  hence  the  total  upward  force  will  be 
2r  X  /  =  2pr;  and  the  tendency  of  this  upward  force  is  to  cause 
the  cylinder  to  give  way  at  A  and  B,  the  upper  part  separating 
from  the  lower. 

This  tendency  is  resisted  by  the  tension  in  the  metal  at  the 
sections  AC  and  BD  ;  hence  at  each  of  these  sections,  there  has 
to  be  resisted  a  tensile  stress  equal  to  \(2pr)  —pr.  This  stress 
is  really  not  distributed  uniformly  throughout  the  cross-section 
of  the  metal  ;  but,  inasmuch  as  the  metal  is  thin,  no  serious 
error  will  be  made  if  it  be  accounted  as  distributed  uniformly. 
The  area  of  each  section,  however,  is  t  X  i  =  /  /  therefore,  if 
T  denote  the  intensity  of  the  tension  in  the  metal  in  a  tangential 
direction  (i.e.,  the  intensity  of  the  hoop  tension),  we  shall  have 


Hence,  to  insure  safety,  T  must  not  be  greater  than  f,  the 
working-strength  of  the  material  for  tension  ;  hence,  putting 


f-pr 
/-•  /, 

we  shall  have 

/  =  7 

as  the  proper  thickness,  when/  =  normal  pressure  per  square 
inch,  and  radius  =  r. 

The  above  are  the  formulas  in  common  use  for  the  deter- 
mination of  the  thickness  of  the  shell  of  a  steam-boiler ;  for  in 
that  case  the  steam-pressure  is  so  great  that  the  tension 
induced  by  any  shocks  that  are  likely  to  occur,  or  by  the  weight 
of  the  boiler,  is  very  small  in  comparison  with  that  induced 
by  the  steam-pressure.  On  the  other  hand,  in  the  case  of  an 
ordinary  water-pipe,  the  reverse  is  the  case. 


RESISTANCE    TO  DIRECT  COMPRESSION. 


To  provide  for  this  case,  Weisbach  directs  us  to  add  to  the 
thickness  we  should  obtain  by  the  above  formulae,  a  constant 
minimum  thickness. 

The  following  -are  his  formulae,  d  being  the  diameter  in 
inches,  /  the  internal  normal  pressure  in  atmospheres,  and  / 
the  thickness  in  inches.  For  tubes  made  of 

Sheet-iron  .........  /*  =  0.00086  pd  -f  0.12 

Cast-iron     .........  t  =  0.00238^  -f-  0.34 

Copper  .......     .     .     .  /  =  o.ooi48/^/  +  0.16 

Lead      ..........  /=  0.00507^+  0.21 

Zinc  ...........  /  =  0.00242  pd  -+-  0.16 

Wood     ..........  /  =  0.03230^  -f-  1.07 

Natural  stone  ........  /=  0.03690/^4-  1.18 

Artificial  stone      .......  t  =  0.05380/^4-  1.58 

§  175.  Resistance  to  Direct  Compression.  —  When  a  piece 
is  subjected  to  compression,  the  distribution  of  the  compressive 
stress  on  any  cross-section  depends,  first,  upon  whether  the 
resultant  of  the  pressure  acts  along  the  line  containing  the  cen- 
tres of  gravity  of  the  sections,  and,  secondly,  upon  the  dimen- 
sions of  the  piece  ;  thus  determining  whether  it  will  bend  or 
not. 

In  the  case  of  an  eccentric  load,  or  of  a  piece  of  such  length 
that  it  yields  by  bending,  the  stress  is  not  uniformly  distributed  ; 
and,  in  order  to  proportion  the  piece,  we  must  determine  the 
greatest  intensity  of  the  stress  upon  it,  and  so  proportion  it 
that  this  shall  be  kept  within  the  working-strength  of  the  ma- 
terial for  compression. 

Either  of  these  cases  is  not  a  case  of  direct  compres- 
sion. 

In  the  case  of  direct  compression  (i.e.,  where  the  stress  over 
each  section  is  uniformly  distributed),  the  intensity  of  the  stress 
is  found  by  dividing  the  total  compression  by  the  area  of  the 


254  APPLIED   MECHANICS. 

section  ;  so  that,  if  P  be  the  total  compression,  and  A  the  area 
of  the  section,  and  /  the  intensity  of  the  compressive  stress, 


On  the  other  hand,  if  f  is  the  compressive  working-strength  oi 
the  material  per  square  inch,  and  A  the  area  of  the  section  in 
square  inches,  then  the  greatest  allowable  load  on  the  piece 
subjected  to  compression  is 


The  same  remarks  as  were  made  in  regard  to  a  suddenly 
applied  load  and  resilience,  in  the  case  of  direct  tension,  apply 
in  the  case  of  direct  compression. 

§  176.  Results  of  Wohler's  Experiments  on  Compressive 
Strength.  —  Wohler  also  made  experiments  in  regard  to  pieces 
subjected  to  alternate  tension  and  compression,  taking,  in  the 
experiments  themselves,  the  case  where  the  metal  is  subjected 
to  alternate  tensions  and  compressions  of  equal  amount. 

The  greatest  stress  of  which  the  piece  would  bear  an  indefi- 
nite number  of  changes  under  these  conditions,  is  called  the 
vibration  safe  strength,  and  is  denoted  by  s. 

Weyrauch  deduces  a  formula  similar  to  that  of  Launhardt 
for  the  greatest  allowable  stress  per  unit  of  area  on  the  piece 
when  it  is  subjected  to  alternate  tensions  and  compressions  of 
different  amounts. 

Thus,  for  Phoenix  Company's  axle  iron,  Wohler  deduces 

/  =  3290  kil.  per  sq.  cent.  =  46800  Ibs.  per  sq.  in., 
u  =  2100  kil.  per  sq.  cent.  =  30000  Ibs.  per  sq.  in., 
s  =  1  1  70  kil.  per  sq.  cent.  =  1  6600  Ibs.  per  sq.  in. 


EXPERIMENTS  ON  COMPRESSIVE  STRENGTH.  2$$ 

Weyrauch's  formula  for  the  ultimate  strength  per  unit  of 
area  is 

{u  —  s     least  maximum  stress    | 
u      greatest  maximum  stress  \ ' 

and,  with  these  values  of  u  and  s,  it  gives 


least  maximum  stress 
a  =    2100 


!i     least  maximum  stress    | 
1    ~  2  greatest  maximum  stress  j      '  per  S(**  cent*» 

11     least  maximum  stress    ) 
i  —  -  —  — = —  -  /•  Ibs.  per  sq.  in. 

2  greatest  maximum  stress ) 


With  a  factor  of  safety  of  3,  we  should  have,  for  the  greatest 
admissible  stress  per  square  inch, 

(          i     least  maximum  stress    ) 
b  =  i oooo <  i — : —         — Jibs. 

(          2  greatest  maximum  stress) 

For  Krupp's  cast-steel, 

/  =  7340  kil.  per  sq.  cent.  =  104400  Ibs.  per  sq.  in., 

u  =  3300  kil.  per  sq.  cent.  =    46900  Ibs.  per  sq.  in.  approximately, 

s  =  2050  kil.  per  sq.  cent.  =     29150  Ibs.  per  sq.  in.  approximately. 

We  have,  therefore,  for  the  breaking-strength  per  unit  of 
area,  according  to  Weyrauch's  formula, 

least  maximum  stress 
a 


or 

a 


!c     least  maximum  stress    ) 
-  il  greatest  maximum  stress  }  kiL  per  Sq'  Cent' 

(  c      least  maximum  stress    ) 

-  469oo| ,  -  fi  greatestmaximumstress[lbs.  per  sq. «.; 


256  APPLIED   MECHANICS. 

and,  using  a  factor  of  safety  of  3,  we  have,  for  the  greatest  admis- 
sible stress  per  square  inch, 

!«r     least  maximum  stress    ) , 
i  —  f:  -  — : —  -  >  Ibs.  per  sq.  in. 

1 1  greatest  maximum  stress  j 


b  —  15630 


The  principles  respecting  an  eccentric  compressive  load,  and 
those  respecting  the  giving-way  of  long  columns  so  far  as  they 
are  known,  can  only  be  treated  after  we  have  studied  the  resist- 
ance of  beams  to  bending;  hence  this  subject  will  be  deferred 
until  that  time. 

EXAMPLES. 

Find  the  proper  working  and  breaking  strength  per  square  inch  to 
be  used  for  a  wrought- iron  rod,  the  extreme  stresses  being  — 

1.  80000  Ibs.  tension  and      6000  Ibs.  compression. 

2.  i  ooooo  Ibs.  tension  and  100000  Ibs.  compression. 

3.  70000  Ibs.  tension  and    60000  Ibs.  compression. 
Do  the  same  for  a  steel  rod. 

§  177.  Resistance  to  Shearing.  —  One  of  the  principal  cases 
where  the  resistance  to  shearing  comes  into  practical  use  is 
that  where  the  members  of  a  structure,  which  are  themselves 
subjected  to  direct  tension  or  compression  or  bending,  are  united 
by  such  pieces  as  bolts,  rivets,  pins,  or  keys,  which  are  sub- 
jected to  shearing.  Sometimes  the  shearing  is  combined  with 
tension  or  with  bending ;  and  whenever  this  is  the  case,  it  is 
necessary  to  take  account  of  this  fact  in  designing  the  pieces. 
It  is  important  that  the  pins,  keys,  etc.,  should  be  equally 
strong  with  the  pieces  they  connect. 

Probably  one  of  the  most  important  modes  of  connection  is 
by  means  of  rivets.  In  order  that  there  may  be  only  a  shearing 
action,  with  but  little  bending  of  the  rivets,  the  latter  must 
fit  very  tightly.  The  manner  in  which  the  riveting  is  done  will 
necessarily  affect  very  essentially  the  strength  of  the  joints; 


RESISTANCE    TO    SHEARING. 


257 


hence  the  only  way  to  discuss  fully  the  strength  of  riveted 
joints  is  to  take  into  account  the  manner  of  effecting  the  rivet- 
ing, and  hence  the  results  of  experiments.  These  will  be 
spoken  of  later  ;  but  the  ordinary  theories  by  which  the  strength 
and  proportions  of  .some  of  the  simplest  forms  of  riveted  joints 
are  determined  will  be  given,  which  theories  are  necessary  also 
in  discussing  the  results  of  experiments  thereon. 

The  principle  on  which  the  theory  is  based,  in  these  simple 
cases,  is  that  of  making  the  resistance  of  the  joint  to  yielding 
equal  in  the  first  three,  and  also  in  either  the  fourth  or  the 
fifth  of  the  ways  in  which  it  is  possible  for  it  to  yield,  as 
enumerated  on  pages  258  and  259. 


A  single-riveted  lap-joint  is  one 
with  a  single  row  of  rivets,  as 
shown  in  Fig.  154. 


A  single-riveted  butt-joint  with 
covering    plate   is   shown   in 

T  C  C 


one 
Fig.  155 


A  single-riveted  butt-joint  with 
two  covering,  plates  is  shown  in 
Fig.  156. 


FIG.  154. 


FIG.  155. 


FIG.  156. 


258 


APPLIED   MECHANICS. 


FIG.  157. 


FIG.  158. 


A  double-riveted  lap-joint  with 
the  rivets  staggered  is  shown  in 
Fig.  157;  one  with  chain  riveting, 
in  Fig.  158. 


Taking  the  case  of  the  single-riveted  lap-joint  shown  in  Fig. 
1 54,  it  may  yield  in  one  of  five  ways :  — 


i°.  By  the  crushing  of  the  plate 
in  front  of  the  rivet  (Fig.  159). 


FIG.  159. 


FIG.  160. 


2°.  By  the  shearing  of  tne  nvet 
(Fig.  160). 


RESISTANCE    TO   SHEARING. 


259 


3°.  By  the  tearing  of  the  plate 
between  the  rivet-holes  (Fig.  161). 

1 

FIG.  161. 


4°.  By  the  rivet  breaking 
through  the  plate  (Fig.  162). 

5°.  By  the  rivet  shearing  out 
the  plate  in  front  of  it. 


Let  us  call 

d  the  diameter  of  a  rivet. 

/  the  pitch  of  the  rivets  ;  i.e.,  FlG-  l62- 

their  distance  apart  from  centre  to  centre. 
t  the  thickness  of  the  plate. 
/    the  lap  of  the  plate  ;  i  e.,  the  distance  from  the  centre 

of  a  rivet-hole  to  the  outer  edge  of  the  plate. 


ft  the  ultimate  tensile  strength  of  the  iron. 
/,  the  ultimate  shearing-strength  of  the  rivet-iron. 
fs>  the  ultimate  shearing-strength  of  the  plate. 
fc  the  ultimate  crushing-strength  of  the  iron. 
We  shall  then  have  — 

i°.   Resistance  of  plate  in  front  of  rivet  to  crushing  =r  fctd. 

2°.   Resistance  of  one  rivet  to  shearing  =  //-  —  Y 

3°.  Resistance  of  plate  between  two  rivet-holes  to  tearing 
=  /«'(/  -  d). 

4°.  Resistance  of  plate  to  being  broken   through  =  a~  , 

d 


where  a  is  a  constant  depending  on  the  material, 
taken  as  empirical  for  the  present. 

A  reasonable  value  of  this  constant  is    /". 


This  may  be 


260<  APPLIED   MECHANICS. 

5°.  Resistance  of  plate  in  front  of  the  rivet  to  shearing 
=  2/,7/. 

Assuming  that  we  know  the  thickness  of  the  plate  to  start 
with,  we  obtain,  by  equating  the  first  two  resistances, 


which  determines  the  diameter  of  the  rivet. 
Equating  3°  and  2°,  we  obtain 


which  gives  the  pitch  of  the  rivets  in  terms  of  the  diameter  of 
the  rivet,  and  the  thickness  of  the  plate. 
Equating,  next,  4°  and  i°,  we  have 


which  gives  the  lap  of  the  plate  needed  in  order  that  it  may  not 
break  through. 

By  equating  5°  and  i°,  we  find  the  lap  needed  that  it  may 
not  shear  out  in  front  of  the  rivet. 

A  similar  method  of  reasoning  would  enable  us  to  determine 
the  corresponding  quantities  in  the  cases  of  double-riveted 
joints,  etc. 

There  are  a  number  of  practical  considerations  which 
modify  more  or  less  the  results  of  such  calculations,  and  which 
can  only  be  determined  experimentally.  A  fuller  account  of 
this  subject  from  an  experimental  point  of  view  will  be  given 
later. 

§  178.  Intensity  of  Stress.  —  Whenever  the  stress  over  a 
plane  area  is  uniformly  distributed,  we  obtain  its  intensity  at 
each  point  by  dividing  the  total  stress  by  the  area  over  which 
it  acts,  thus  obtaining  the  amount  per  unit  of  area.  When,  how- 
ever, the  stress  is  not  uniformly  distributed,  or  when  its  inten- 


INTENSITY  OF  STRESS. 


26l 


sity  varies  at  different  points,  we  must  adopt  a  somewhat  differ- 
ent definition  of  its  intensity  at  a  given  point.  In  that  case,  if 
we  assume  a  small  area  containing  that  point,  and  divide  the 
stress  which  acts  on  that  area  by  the  area,  we  shall  have,  in  the 
quotient,  an  approximation  to  the  intensity  required,  which  will 
approach  nearer  and  nearer  to  the  true  value  of  the  intensity  at 
that  point,  the  smaller  the  area  is  taken. 

Hence  the  intensity  of  a  variable  stress  at  a  given  point  is,  -- 

The  limit  of  the  ratio  of  the  stress  acting  on  a  small  area 
containing  that  point,  to  the  area,  as  the  latter  grows  smaller  and 
smaller. 

By  dividing  the  total  stress  acting  on  a  certain  area  by  the 
entire  area,  we  obtain  the  mean  intensity  of  the  stress  for  the 
entire  area. 

§  179.    Graphical  Representation  of  Stress — A  conven- 
ient  mode    of   representing   stress 
graphically  is  the  following:  — 

Let  AB  (Fig.  163)  be  the  plane 
surface  upon  which  the  stress  acts  ; 
let  the  axes  OX  and  OY  be  taken 
in  this  plane,  the  axis  OZ  being  at 
right  angles  to  the  plane. 

Conceive  a  portion  of  a  cylinder 
whose  elements  are  all  parallel  to 
OZ,  bounded  at  one  end  by  the 
given  plane  surface,  and  at  the 
other  by  a  surface  whose  ordinate 
many  units  of  length  as  there  are  units  of  force  in  the  intensity 
of  the  stress  at  that  point  of  the  given  plane  surface  where  the 
ordinate  cuts  it. 

The  volume  of  such  a  figure  will  evidently  be 

V  =  ffzdxdy  =  ffpdxdy, 
where  z  —  /  =  intensity  of  the  stress  at  the  given  point. 


FIG.  163. 

at  any  point   contains  as 


262 


APPLIED   MECHANICS. 


Hence  the  volume  of  the  cylindrical  figure  will  contain  as 
many  units  of  volume  as  the  total  stress  contains  units  of 
force  ;  or,  in  other  words,  the  total  stress  will  be  correctly  repre- 
sented by  the  volume  of  the  body. 

If  the  stress  on  the  plane 
figure  is  partly  tension  and 
partly  compression,  the  sur- 
face whose  ordinates  repre- 
sent the  intensity  of  the 
stress  will  lie  partly  on  one 
side  of  the  given  plane  sur- 
face and  partly  on  the  other  ; 
this  surface  and  the  plane 
surface  on  which  the  stress 
acts,  cutting  each  other  in 
some  line,  straight  or  curved, 
as  shown  in  Fig.  164.  In  that 


FlG- 


case,  the  magnitude  of  the  resultant  stress  P  —  V  — 

will  be  equal  to  the  difference  of  the  wedge-shaped  volumes 

shown  in  the  figure. 

It  will  be  observed  that  the  above  method  of  representing 
stress  graphically  represents,  i°,  the  intensity  at  each  point  of 
the  surface  to  which  it  is  applied  ;  and,  2°,  the  total  amount 
of  the  stress  on  the  surface.  It  does  not,  however,  represent 
its  direction,  except  in  the  case  when  the  stress  is  normal  to 
the  surface  on  which  it  acts. 

In  this  latter  case,  however,  this  is  a  complete  representa- 
tion of  the  stress. 

The  two  most  common  cases  of  stress  are,  i°,  uniform  stress, 
and,  2°,  uniformly  varying  stress.  These  two  cases  are  repre- 
sented respectively  in  Figs.  165  and  166;  the  direction  also 
being  correctly  represented  when,  as  is  most  frequently  the 
case,  the  stress  is  normal  to  the  surface  of  action.  In  Fig. 
165,  AB  is  supposed  to  be  the  surface  on  which  the  stress 


GRAPHICAL   REPRESENTATION  OF  STRESS. 


263 


acts ;  the  stress  is  supposed  to  be  uniform,  and  normal  to  the 

surface  on  which  it  acts ;  the  bound- 
ing surface   in  this  case  becomes  a 

plane  parallel  to  AB ;   the  intensity 

of  the  stress  at  any  point,  as  P,  will 

be   represented    by  PQ;    while    the 

whole  cylinder  will  contain  as  many 

units  of  volume  as  there  are  units  of 

force  in  the  whole  stress. 

Fig.  i(56  represents   a   uniformly 

varying  stress.     Here,  again,  AB  is 

the  surface  of  action,  and  the  stress 

is  supposed  to  vary  at  a  uniform  rate  FlG>  l65' 

from  the  axis  O  Y.  The  upper  bounding  surface  of  the  cylin- 
drical figure  which  represents  the  stress 
becomes  a  plane  inclined  to  the  XOY 
plane,  and  containing  the  axis  O  Y. 

In  this  case,  if   a   represent  the   in- 
tensity of  the  stress  at  a  unit's  distance 
°  from  O  Y,  the  stress  at  a  distance  x  from 
OY  will  be/  =  ax,  and  the  total  amount 
of  the  stress  will  be 


FIG.  166. 


P  =  ffpdxdy  =  affxdxdy. 


When  a  stress  is  oblique  to  the  surface  of  action,  it  may  be 
represented  correctly  in  all  particulars,  except  in  direction,  in 
the  above-stated  way. 

§  1 80.  Centre  of  Stress.  — The  centre  of  stress,  or  the 
point  of  the  surface  at  which  the  resultant  of  the  stress  acts, 
often  becomes  a  matter  of  practical  importance.  If,  for  con- 
venience, we  employ  a  system  of  rectangular  co-ordinate  axes, 
of  which  the  axes  OX  and  OY  are  taken  in  the  plane  of  the 
surface  on  which  the  stress  acts,  and  if  we  let  p  =  $(x,  y)  be 


264  APPLIED   MECHANICS. 

the  intensity  of  the  stress  at  the  point  (x,  y),  we  shall  have, 
for  the  co-ordinates  of  the  centre  of  stress, 

ffxpdxdy  J'Sypdxdy 

: 


(see  §  42),  where  the  denominator,  or  ffpdxdy,  represents  the 
total  amount  of  the  stress. 

When  the  stress  is  positive  and  negative  at  different  parts 
of  the  surface,  as  in  Fig.  164,  the  case  may  arise  when  the  posi- 
tive and  negative  parts  balance  each  other,  and  hence  the 
stress  on  the  surface  constitutes  a  statical  couple.  In  that  case 

Sfpdxdy  =  o. 

§  181.  Uniform  Stress.  —  In  the  case  of  uniform  stress,  we 
have  — 

i°.  The  intensity  of  the  stress  is  constant,  or  /  =  a  con- 
stant. 

2°.  The  volume  which  represents  it  graphically  becomes  a 
cylinder  with  parallel  and  equal  bases,  as  in  Fig.  165. 

3°.  The  centre  of  stress  is  at  the  centre  of  gravity  of  the 
surface  of  action  ;  for  the  formulae  become,  when  /  is  constant, 

_  pffxdxdy  _  ffxdxdy  _ 
Xl  ~~~  pffdxdy  ~~~'  ~~ 

pffydxdy 
= 


pffdxdy  ~'    ffdxdy          °' 

"where  x0,  y0,  are  the  co-ordinates  of  the  centre  of  gravity  of  the 
surface. 

Examples  of  uniform  stress  have  already  been  given  in  the 
cases  of  direct  tension,  direct  compression,  and,  in  the  case  of 
riveted  joints,  for  the  shearing-force  on  the  rivet. 


UNIFORMLY   VARYING   STRESS.  26$ 

§182.  Uniformly  Varying  Stress.  —  Uniformly  varying 
stress  has  already  been  denned  as  a  stress  whose  intensity  varies 
uniformly  from  a  given  line  in  its  own  plane  ;  and  this  line  will 
be  called  the  Neutral  Axis.  Thus,  if  the  plane  be  taken  as  the 
XOY  plane  (Fig.  166),  and  the  given  line  be  taken  as  OY,  we 
shall  have,  if  a  denotes  the  intensity  of  the  stress  at  a  unit's 
distance  from  OY,  and  x  the  distance  of  any  special  point  from 
O  Y,  that  the  intensity  of  the  stress  at  the  point  will  be 

p  =  ax. 

The  total  amount  of  the  stress  will  be 

P=  affxdxdy. 

The  total  moment  of  the  stress  about  O  Y  will  be  found  by 
multiplying  each  elementary  stress  by  its  leverage.  This  lever- 
age is,  in  the  case  of  normal  stress,  x  ;  hence  in  that  case  the 
moment  of  any  single  elementary  force  will  be 


and  the  total  moment  of  the  stress  will  be 

M  -  affx^dxdy  =  al. 

In  the  case  of  oblique  stresxs,  this  result  has  to  be  modified, 
as  the  leverage  is  no  longer  x.  Confining  ourselves  to  stress 
normal  to  the  plane  of  action,  we  have,  for  the  co-ordinates  of 
the  centre  of  stress, 

_  ffpxdxdy  _  affx*dxdy  _  ffx^dxdy  _  ffx*dxdy_    I 
ffpdxdy   ~~  P  =  ffxdxdy    ''          xQA      ~  x0A 

_  ffpydxdy  _  affxydxdy  _  ffxydxdy  _  ffxydxdy 
~~  ffpdxdy  ~~  P  ~~  ffxdxdy   =         x*A 

since 

P  =  affxdxdy  =  aXoA, 

where  xm  ym  are  the  co-ordinates  of  the  centre  of  gravity,  and 
A  is  the  area  of  the  surface  of  action. 


266  APPLIED   MECHANICS. 


§  183.  Case  of  a  Uniformly  Varying  Stress  which 
amounts  to  a  Statical  Couple.  —  Whenever  P  =  o,  we  have 

affxdxdy  =  o       /.    ffxdxdy  =  o       .'.    x^A  =  o       .*.    x0  =  o. 

In  this  case,  therefore,  we  have  — 

i°.  There  is  no  resultant  stress,  and  hence  the  whole  stress 
amounts  to  a  statical  couple. 

2°.  Since  XQ  =  o,  the  centre  of  gravity  of  the  surface  of 
action  is  on  the  axis  OY,  which  is  the  neutral  axis. 

Hence  follows  the  proposition  :  — 

When  a  uniformly  varying  stress  amounts  to  a  statical  couple, 
the  neutral  axis  contains  (passes  through)  the  centre  of  gravity 
of  the  surface  of  action. 

In  this  case  there  is  no  .single  resultant  of  the  stress ;  but 
the  moment  of  the  couple  will  be,  as  has  been  already  shown, 

M  =  affx2dxdy. 

§  184.  Example  of  Uniformly  Varying  Stress.  —  One  of 
the  most  common  examples  of  uniformly  varying  stress  is  that 
of  the  pressure  of  water  upon  the  sides  of  the  vessel  contain- 
ing it. 

Thus,  let  Fig.  167  represent  the  vertical  cross-section  of  a 
reservoir  wall,  the  water  pressing  against  the 
vertical  face  AB.  It  is  a  fact  established  by 
experiment,  that  the  intensity  of  the  pressure 
of  any  body  of  water  at  any  point  is  propor- 
tional to  the  depth  of  the  point  below  the 
free  upper  level  of  the  water,  and  normal  to 
the  surface  pressed  upon.  Hence,  if  we  sup- 
pose the  free  upper  level  of  the  water  to  be 
even  with  the  top  of  the  wall,  the  intensity 
of  the  pressure  there  will  be  zero  ;  and  if  we  represent  by  CB 
the  intensity  of  the  pressure  at  the  bottom,  then,  joining^  and 


STRESSES  IN  BEAMS   UNDER    TRANSVERSE  LOAD.      267 


C  we  shall  have  the  intensity  of  the  pressure  at  any  point,  as 
Dt  represented  by  ED,  where 

ED  :  CB  =  AD  :  AB. 

Here,  then,  we  have  a  case  of  uniformly  varying  stress  nor- 
mal to  the  surface  on  which  it  acts. 

§  185.  Fundamental  Principles  of  the  Common  Theory 
of  the  Stresses  in  Beams  under 
a  Transverse  Load.  —  Fig.  168 
shows  a  beam  fixed  at  one  end  and 
loaded  at  the  other,  while  Fig.  169 
shows  a  beam  supported  at  the 
ends  and  loaded  at  the  middle. 
Let,  in  each  case,  the  plane  of  the 
paper  contain  a  vertical  longi- 
tudinal section  of  the  beam.  In 

Fig.  1 68, 
it  is  evi- 
dent that 
the  upper 

fibres  are  lengthened,  while  the  lower 
ones  are  shortened,  and  vice  versa  in 
Fig.    169.      In    either   case,    there    is, 
somewhere    between    the    upper   and 
lower  fibres,  a  fibre  which  is  neither 
elongated  nor   com- 
pressed. 

Let     CN    repre- 
sent that  fibre,  Fig. 
.168,    and    CP,    Fig. 
169.     This  line  may 
be  called  the  neutral 
FIG.  i6g.  line  of   the  longitu- 

dinal section ;  and,  if  a  section  be  made  at  any  point  at  right 


268  APPLIED   MECHANICS. 

angles  to  this  line,  the  horizontal  line  which  lies  in  the  cross- 
section,  and  cuts  the  neutral  lines  of  all  the  longitudinal  sec- 
tions, or,  in  other  words,  the  locus  of  the  points  where  the 
neutral  lines  of  the  longitudinal  sections  cut  the  cross-section, 
is  called  the  Neutral  Axis  of  the  cross-section.  In  the  ordinary 
theory  of  the  stresses  in  beams,  a  number  of  assumptions  are 
made,  which  will  now  be  enumerated. 


ASSUMPTIONS    MADE    IN    THE    COMMON    THEORY    OF    BEAMS. 

ASSUMPTION  No.  I.  —  If,  when  a  beam  is  not  loaded,  a 
plane  cross-section  be  made,  this  cross-section  will  still  be  a 
plane  after  the  load  is  put  on,  and  bending  takes  place.  From 
this  assumption,  we  deduce,  as  a  consequence,  that,  if  a  certain 
cross-section  be  assumed,  the  elongation  or  shortening  per  unit 
of  length  of  any  fibre  at  the  point  where  it  cuts  this  cross-sec- 
tion, is  proportional  to  the  distance  of  the  fibre  from  the  neutral 
axis  of  the  cross-section. 

Proof.  —  Imagine  two  originally  parallel  cross-sections  so 
near  to  each  other  that  the  curve  in  which  that  part  of  the 
neutral  line  between  them  bends  may,  without  appreciable  error, 
be  accounted  circular.  Let  ED  and  GH  (Fig.  168  or  Fig.  169) 
be  the  lines  in  which  these  cross-sections  cut  the  plane  of  the 
paper,  and  let  O  be  the  point  of  intersection  of  the  lines  ED 
and  GH.  Let  OF  =  r,  FL  =  y,  FK  =  /,  LM  =  /  +  a/,  in 
which  a  is  the  strain  or  elongation  per  unit  of  length  of  a  fibre 
at  a  distance  y  from  the  neutral  line,  y  being  a  variable  ;  then, 
because  FK  and  LM  are  concentric  arcs  subtending  the  same 
angle  at  the  centre,  we  shall  have  the  proportion 

r  +  y       I  -\-  ol  y 

—^  =  -y-     or     i  +  a  =  i  +  £ 

y 

.'.a  =  -  Or     a  = 

r 


ASSUMPTIONS  IN   THE    COMMON  THEORY  OF  BEAMS.   269 


but  as  y  varies  for  different  points  in  any  given  cross-section, 
while  r  remains  the  same  for  the  same  section,  it  follows,  that, 
if  a  certain  cross-section  be  assumed,  the  strain  of  any  fibre  at 
the  point  where  it  cuts  this  cross-section  is  proportional  directly 
to  the  distance  of  this  fibre  from  the  neutral  axis  of  the  cross- 
section. 

ASSUMPTION  No.  2.  —  This  assumption  is  that  commonly 
known  as  Hooke  s  Law.  It  is  as  follows  :  "  Ut  tensio  sic  vis  ;  " 
i.e.,  The  stress  is  proportional  to  the  strain,  or  to  the  elonga- 
tion or  compression  per  unit  of  length.  As  to  the  evidence  in 
favor  of  this  law,  experiment  shows,  that,  as  long  as  the  mate- 
rial is  not  strained  beyond  safe  limits,  this  law  holds.  Hence, 
making  these  two  assumptions,  we  shall  have  :  At  a  given 
cross-section  of  a  loaded  beam,  the  direct  stress  on  any  fibre 
varies  directly  as  the  distance  of  the  fibre  from  the  neutral  axis. 
Hence  it  is  a  uniformly  varying  stress,  and  we  may  repre- 
sent it  graphically  as  follows :  Let 
ABCD,  Fig.  170,  be  the  cross-sec- 
tion of  a  beam,  and  KL  the  neutral 
axis.  Assume  this  for  axis  OY,  and 
draw  the  other  two  axes,  as  in  the 
figure.  If,  now,  EA  be  drawn  to 
represent  the  intensity  of  the  direct 
(normal)  stress  at  A,  then  will  the 
pair  of  wedges  AEFBKL  and 
DCHGKL  represent  the  stress  graphically,  since  it  is  uni- 
formly varying. 

POSITION   OF   NEUTRAL   AXIS. 

ASSUMPTION  No.  3. — It  will  next  be  shown  that,  on  the 
two  assumptions  made  above,  and  from  the  further  assumption 
that  the  deformation  of  each  fibre  of  the  beam  parallel  to  its* 
longitudinal  axis  is  due  to  the  forces  acting  on  its  ends 


FIG.  170. 


2~0  APPLIED    MECHANICS. 

and  that  it  suffers  no  traction  from  neighboring  fibres,  it  fol- 
lows that  the  neutral  axis  must  pass  through  the  centre  of 
gravity  of  the  cross-section. 


»               D 

N                                                                  0 

1                                    1 

|IC               E 

1  AA            iff  —  *£  ""vi  B' 

FIG.  171.  FIG.  172. 

Since  the  curvatures  in  Figs.  168  and  169  are  exaggerated 
in  order  to  render  them  visible,  Figs.  171  and  172  have  been 
drawn.  If,  now,  we  assume  a  section  DE,  such  that  AD  =  x 
(Fig.  171)  and  NE  =  x  (Fig.  172),  and  consider  all  the  forces 
acting  on  that  part  of  the  beam  which  lies  to  the  right  of  DE 
(i.e.,  both  the  external  forces  and  the  stresses  which  the  other 
parts  of  the  beam  exert  on  this  part),  we  must  find  them  in 
equilibrium.  The  external  forces  are,  in  Fig.  172,  — 

i°.  The  loads  acting  between  B  and  E ;  in  this  case  there 
are  none. 

2°.  The  supporting  force  at  B ;  in  this  case  it  is  equal  to 

W 

— ,  and  acts  vertically  upwards. 

In  Fig.  171  they  are, — 

The  loads  between  D  and  N ' ;  in  this  case  there  is  only  the 
one,  W  at  N. 

The  internal  forces  are  merely  the  stresses  exerted  by  the 
other  parts  of  the  beam  on  this  part :  they  are,  — 

i°.  The  resistance  to  shearing  at  the  section,  which  is  a 
vertical  stress. 

2°.  The  direct  stresses,  which  are  horizontal. 

Now,  since  the  part  of  the  beam  to  the  right  of  DE  is  at 
rest,  the  forces  acting  on  it  must  be  in  equilibrium ;  and,  since 


POSITION  OF  NEUTRAL   AXIS.  27l 

they  are  all  parallel  to  the  plane  of  the  paper,  we  must  have 
the  three  following  conditions  ;  viz.,  — 

i°.  The  algebraic  sum  of  the  vertical  forces  must  be  zero. 

2°.  The  algebraic  sum  of  the  horizontal  forces  must  be  zero. 

3°.  The  algebraic  sum  of  the  moments  of  the  forces  about 
any  axis  perpendicular  to  the  plane  of  the  paper  must  be 
zero. 

But,  on  the  above  assumptions,  the  only  horizontal  forces 
are  the  direct  stresses  :  hence  the  algebraic  sum  of  these  direct 
stresses  must  be  zero  ;  or,  in  other  words,  the  direct  stresses 
must  be  equivalent  to  a  statical  couple. 

Now,  it  has  already  been  shown,  that,  whenever  a  uniformly 
varying  stress  amounts  to  a  statical  couple,  the  neutral  axis 
must  pass  through  the  centre  of  gravity  of  the  surface  acted 
upon.  Hence  in  a  loaded  beam,  if  the  three  preceding  assump- 
tions be  made,  it  follows  that  the  neutral  axis  of  any  cross- 
section  must  contain  the  centre  of  gravity  of  that  section. 

By  way  of  experimental  proof  of  this  conclusion,  Barlow 
has  shown  by  experiment,  that,  in  a  cast-iron  beam  of  rectangu- 
lar section,  the  neutral  axis  does  pass  through  the  centre  of 
gravity  of  the  section. 

RESUME. 

The  conclusions  arrived  at  from  the  foregoing  are  as  fol- 
lows :  — 

i°.  That  at  any  section  of  a  loaded  beam,  if  a  horizontal 
line  be  drawn  through  the  centre  of  gravity  of  the  section, 
then  the  fibres  lying  along  this  line  will  be  subjected  neither 
to  tension  nor  to  compression  ;  in  other  words,  this  line  will  be 
the  neutral  axis  of  the  section. 

2°.  The  fibres  on  one  side  of  this  line  will  be  subjected  to 
tension,  those  on  the  other  side  being  subjected  to  compres- 
sion ;  the  tension  or  compression  of  any  one  fibre  being  proper 
tionai  to  its  distance  from  the  neutral  axis. 


2/2  APPLIED    MECHANICS. 

The  first  of  the  three  assumptions  of  the  common  theory 
was  not  accepted  by  St.  Venant,  who  developed  by  means  of 
the  methods  of  the  Theory  of  Elasticity  a  theory  of  beams 
based  upon  the  second  and  third  assumptions  only.  A  study 
of. St.  Venant's  theory  involves,  however,  far  more  complica- 
tion, and  requires  a  good  previous  knowledge  of  the  Theory  of 
Elasticity.  Moreover  the  results  of  the  two  theories  as  far  as 
the  determination  of  the  outside  fibre-stresses  and  of  the  de- 
flections are  practically  in  agreement,  while,  on  the  other  hand, 
the  intensities  of  the  shearing-forces  as  computed  by  the  two 
theories  are  not  in  agreement. 

The  St.  Venant  theory  may  be  found  in  several  treatises 
upon  the  Theory  of  Elasticity. 

§  1 86.  Shearing-Force  and  Bending-Moment.  —  In  deter- 
mining the  strength  of  a  beam,  or  the  proper  dimensions  of  a 
beam  to  bear  a  certain  load,  when  we  assume  the  neutral  axis 
to  pass  through  the  centre  of  gravity  of  the  cross-section,  we 
have  imposed  the  second  of  the  three  last-mentioned  conditions 
of  equilibrium.  The  remaining  two  conditions  may  otherwise 
be  stated  as  follows  :  — 

i°.  The  total  force  tending  to  cause  that  part  of  the  beam 
that  lies  to  one  side  of  the  section  to  slide  by  the  other  part, 
must  be  balanced  by  the  resistance  of  the  beam  to  shearing  at 
the  section. 

2°.  The  resultant  moment  of  the  external  forces  acting  on 
that  part  of  the  beam  that  lies  to  one  side  of  the  section,  about 
a  horizontal  axis  in  the  plane  of  the  section,  must  be  balanced 
by  the  moment  of  the  couple  formed  by  the  resisting  stresses. 

The  shearing-force  at  any  section  is  the  force  with  which  the 
part  of  the  beam  on  one  side  of  the  section  tends  to  slide  by  the 
part  on  the  other  side.  In  a  beam  free  at  one  end,  it  is  equal  to 
the  sum  of  the  loads  between  the  section  and  the  free  end.  In 
a  beam  supported  at  both  ends,  it  is  equal  in  magnitude  to  the 
difference  between  the  supporting  force  at  either  end,  'and 
the  sum  of  the  loads  between  the  section  and  that  support. 


SHEARING-FORCE  AND   BEND  ING-MOMENT.  2?  3 

The  bending-moment  at  any  section  is  the  resultant  moment 
of  the  external  forces  acting  on  the  part  of  the  beam  to  one 
side  of  the  section,  these  moments  being  taken  about  a  hori- 
zontal axis  in  the  section. 

In  a  beam  free  at  one  end,  it  is  equal  to  the  sum  of  the 
moments  of  the  loads  between  the  section  and  the  free  end, 
about  a  horizontal  axis  in  the  section. 

In  a  beam  supported  at  both  ends,  it  is  the  difference  be- 
tween the  moment  of  either  supporting  force,  and  the  sum  of 
the  moments  of  the  loads  between  the  section  and  that  sup- 
port ;  all  the  moments  being  taken  about  a  horizontal  axis  in 
the  section. 

Hence  the  two  conditions  of  equilibrium  may  be  more 
briefly  stated  as  follows  :  — 

i°.  The  shearing-force  at  the  section  must  be  balanced 
by  the  resistance  opposed  by  the  beam  to  shearing  at  the 
section. 

2°.  The  bending-moment  at  the  section  must  be  balanced 
by  the  moment  of  the  couple  formed  by  the  resisting  stresses. 

It  is  necessary,  therefore,  in  determining  the  strength  of  a 
beam,  to  be  able  to  determine  the  shearing-force  and  bending- 
moment  at  any  point,  and  also  the  greatest  shearing-force  and 
the  greatest  bending-moment,  whatever  be  the  loads. 

A  table  of  these  values  for  a  number  of  ordinary  cases  will 
now  be  given  ;  but  I  should  recommend  that  the  table  be  merely 
considered  as  a  set  of  examples,  and  that  the  rules  already 
given  for  finding  them  be  followed  in  each  individual  case. 

Let,  in  each  case,  the  length  of  the  beam  be  /,  and  the 
total  load  W.  When  the  beam  is  fixed  at  one  end  and  free  at 
the  other,  let  the  origin  be  taken  at  the  fixed  end ;  when  it  is 
supported  at  both  ends,  let  it  be  taken  directly  over  one  support. 
Let  x  be  the  distance  of  any  section  from  the  origin.  Then  we 
shall  have  the  results  given  in  the  following  table : — 


274 


APPLIED   MECHANICS. 


At  Dista 
from  O 


ifeh. 


ft 


Distance 
m  Origin. 


' 
« 


«" 

pq 


Si 


I 


T?       & 

g  '-3 

>, 


IS 


11 


4'g 
«! 
It 


W      iJ 


.g 


r-    4> 

u 


PQ 


MOMENTS   OF  INERTIA    OF  SECTIONS.  2/5 

In  a  beam  fixed  at  one  end  and  free  at  the  other,  the  great- 
est shearing-force,  and  also  the  greatest  bending-moment,  are  at 
the  fixed  end.  In  a  beam  supported  at  both  ends,  and  loaded 
at  the  middle,  or  with  a  uniformly  distributed  load,  the  greatest 
shearing-force  is  at  either  support,  the  greatest  bending-moment 
being  at  the  middle.  In  the  last  case  (i.e.,  that  of  a  beam  sup- 
ported at  the  ends,-  and  having  a  single  load  not  at  the  middle),. 
the  greatest  bending-moment  is  at  the  load ;  the  greatest  shear- 
ing-force being  at  that  support  where  the  supporting  force  is 
greatest. 

§  187.  Moments  of  Inertia  of  Sections.  —  In  the  usual 
methods  of  determining  the  strength  of  a  beam  or  column,  it 
is  necessary  to  know,  i°,  the  distance  from  the  neutral  axis  of 
the  section  to  the  most  strained  fibres ;  2°,  the  moment  of  in- 
ertia of  the  section  about  the  neutral  axis.  The  manner  of 
finding  the  moments  of  inertia  has  been  explained  in  Chap.  II. 

In  the  following  table  are  given  the  areas  of  a  large  number 
of  sections,  and  also  their  moments  of  inertia  about  the  neutral 
axis,  which  is  the  axis  YY  in  each  case.  These  results  should 
be  deduced  by  the  student. 


276 


APPLIED  MECHANICS. 


Distance  of  YY  from 
Extreme  Fibres. 


S. 
HP 


|    1!    II 


MOMENTS  OF  INERTIA    OF  SECTIONS. 


277 


fa  ^r 

S-     SS 


5ia  5lx 


13 
w    rt 


£  -o 


CQ 


278 


APPLIED   MECHANICS. 


o  a 

§1 

rt  X 
w  W 
Q 


vO 


f 


I       I 


0  -S 

-8     S 


3  ^ 

^H  ^ 


>     MH 

55     0 

"5 


•g  §  °-s  § 

^     bfl   tfl   ^     bo 


!* 


MOMENTS  OF  INERTIA    OF  SECTIONS. 


'279 


+  + 


4- 

^  IN 


•*?!« 


II- ^  I 

H       ^ 


c/T    > 

|  -S'S 

4 «  s,-^ 

's  ^  «  s 

N  2«§ 

"§•  s  •*  « 

£*     K 


if  fi  ii  + 


•S   o 


<  H 


o 

'^3     rt  oS  rt 

Vj      (U  (L)  0 

•o    i-  >_  i- 


JA, 


^  —  U 


280 


APPLIED   MECHANICS. 


o   g 
4>    c 


SJIN 

u 


II 


•**   rj- 

fp 


1 


•8" 


V     V. 
II      II 


i2  « 
x  »a 
«J5 

131 

' 


11 


I     « 

'~    rt 


i 


£ 


MOMENTS   OF  INERTIA    OF  SECTIONS. 


28l 


1  -^1 

^      8 

°  8  u 

•3       1 

V* 

«-   1  «  . 

•0 

•-  £  1* 

*  « 

^  "5    <u 

°       i; 

-G              "-Q 

rt          K 

~^     § 

s 

°     w     £ 

f         1 

T5        «        yj 

<U     G     nj 

0 

w    ^J      o 

o    ^ 

III 

C 

•s 

!*l 

rt     bJO   c/3 

-' 

I         1          ^          fi 

c     ^ 

'1  ^ 

£  j^  o 

+       I' 

(J^       ^.|     ^J 

1|l 

8   S  -g 

1 

"» 

<U    i^     O 

N 

sli 

^ 

^                   ^                     ^                          I, 

O    ^^"^      CT3 

||)              g|)               |  |            ||||^ 

c/3  '53    ^ 

^J        ^J         ^^      J  *5"^  s  ^ 

n     G     3 

g 

^v                ^^                 ^w            to    2  ^  —   "^ 

0    -2    ""* 

'ft 

oj^i                 4)^i<u            t>rt            ^-t->         ^3'^g 

<U     O     G 

-G     o,^ 

4-> 

1 

~  .2  "§       '•?:»!       '-=  «         §  ^  ^  |  ^  f 

Xrt<U           ^rtS            ^"3^         c3           2i          "P0 

to       <u         w       „.          Mo''tt'3.2'i'aow 
"Sec        'Sex         'J3-Qo      o'xuc^w 

j-jrtO           _j,j    rt    *-             j-.rt*.       ^rtortofi 

"rt  *hO  ^ 

C/2                         C/2                          C/2                     U 

>>    G 

O                            "->                             N                                  ro 

bJO    C   JU 

2"                2"                2s                   2" 

•^     O     C4 

jS>- 

•—  <     crj    *"O 

»o>-              \                          ^« 

'o    ^    c 

w 

<---i                   \                   m                                 ?            ^' 

s  §-r>. 

<U     O     00    ^» 

1 

V       i             x  fe%x 

.X3    J3     «   .t3 

;t-|| 

J-    L-N  "FT 

^  "5  ^o 

>fo 

282 


APPLIED   MECHANICS. 


.5      + 


C      <L>    T3 
rt    X!     fi 

it!! 

^    P    S    <u 


0     .        M_ 


§  «  £  S 

•M     U)    rt     e 

3   c   a  S 

.g     rt    T3     ° 

^1     ^ 

oq  .5P    -  *- 


u  ,e  <u 

— <  bfl  H 

O  13  C 

1-.  O  nj 


. 

0 


.22   - 


MOMENTS   OF  INERTIA    OF  SECTIONS. 


283 


0  2  a 

aj    **-i     w 
O  V 


k|M 


CT1 


Cfl  M 

O 

CJ 


Sh 
II 
fe 


! 

ii 

I 


5 


st: 


VM        " 

O     fi 

si  S 


3      ^ 


284 


APPLIED   MECHANICS. 


o  o   D 

u  H*  a 

u  <u 

a  is 

rt  x 

S5  W 
Q 


+ 
•^ 


I 


•». 


^    k 
w    S 

"i  '-3 


1  <    1 


3     C    42 

- 


MOMENTS   OF  INERTIA    OF  SECTIONS. 


285 


.<«     a 

K  JO 


0  °  a 

rl 


IN    s»  |  N       s»  I  N    <i  I ( 
II  II  II  H 


ripti 


-1  1    -1 

z  I        Egg 

<  U  J  <  U 


in 


Oq 


•?< 


Q        CQ 


>•      < 


:i 


286 


APPLIED   MECHANICS. 


§  188.  Cross-Sections  of  Phoenix  Columns  considered 
as  made  of  Lines.  —  It  is  to  be  observed  that  the  moments 
of  inertia  are  the  same  for  all  axes  passing  through  the  centre. 
Thickness  —  /,  radius  of  round  ones  =  r,  area  of  each  flange 
=  a,  length  of  each  flange  =  /. 


Figure. 


Description. 


A. 


Y2II 


Y2I2 


Four  flanges 


2-nrt  + 


20. 


Eight  flanges 


2irrt  +  8a 


(-0° 


Square,  four  flanges, 
r  —  radius  of  cir- 
cumscribed circle 


Six  flanges 


6a 


REPRESENTATION  OF  B ENDING-MOMENTS. 


287 


§  189.   Graphical  Representation  of  Bending-Moments. — 

The  bending-moment  at  each  point  of  a  loaded  beam  may  be 
represented  graphically  by  lines  laid  off  to  scale,  as  will  be 
shown  by  examples. 

I.  Suppose  we  have  the  cantilever  shown  in  Fig.  215, 
loaded  at  D  with  a  load  W ':  then 
will  the  bending-moment  at  any 
section,  as  at  Ft  be  obtained  by 
multiplying  W  by  FD ;  that  at  AC 
being  W  X  (AB).  If,  now,  we  lay 
off  CE  to  scale  to  represent  this, 
i.e.,  having  as  many  units  of  length 


FIG.  215. 


as  there  are  units  of  moment  in  the  product  W  X  (AB),  and 
join  E  with  D,  then  will  the  ordinate  FG  of  any  point,  as  G, 
represent  (to  the  same  scale)  the  bending-moment  at  a  section 
through  F. 

II.  If  we  have  a  uniformly  distributed  load,  we  should  have, 
for  the  line  corresponding  to  CE  in  Fig.  215,  a  curve.  This  is 

shown  in  Fig.  216,  where  we  have  the 
uniformly  distributed  load  EIGF.  If 
we  take  the  origin  at  D,  as  before, 
we  have,  for  the  bending-moment,  at  a 
distance  ^  from  the  origin,  as  has  been 

W 

shown,  — -(/  —  xY  ;  and  by  giving  x  dif- 
ferent values,  and  laying  off  the  corresponding  value  of  the 
bending-moment,  we  obtain  the  curve  CA,  any  ordinate  of 
which  will  represent  the  bending-moment  at  the  corresponding 
point  of  the  beam. 

When  we  have  more  than  one  load  on  a  beam,  we  must  draw 
the  curve  of  bending-moments  for  each  load  separately,  and 
then  find  the  actual  bending-moment  at  any  point  of  the  beam 


FIG.  216. 


283 


APPLIED  MECHANICS. 


by  taking  the  sum  of  the  ordinates  (drawn  from  that  point)  of 
each  of  these  separate  curves  or  straight  lines.  If  we  then 
draw  a  new  curve,  whose  ordinates  are  these  sums,  we  shall  have 
the  actual  curve  of  bending-moments  for  the  beam  as  loaded. 
Some  examples  will  now  be  given,  which  will  explain  them- 
selves. 

III.  Fig.  217  shows  a  cantilever  with  three  concentrated 
loads.  The  line  of  bending-moments 
for  the  load  at  C  is  CE,  that  for  the 
load  at  O  is  OF,  and  for  the  load  at  P 
is  PG.  They  are  combined  above  the 
beam  by  laying  off  AH  —  DE,  HK  = 
DF,  and  KL  =  DG,  and  thus  obtaining 
the  broken  line  LMNB,  which  is  the 
line  of  bending-moments  of  the  beam 
loaded  with  all  three  loads. 


FIG.  217. 

IV.  Fig.  218  shows  the  case  of  a  beam  supported  at  both 
ends,   and   loaded  at  a   single .  point 

D;  ALB  is  the  line  of  bending- 
moments  when  the  weight  of  the 
beam  is  disregarded,  so  that  xy  = 
bending-moment  at  x.  FIG.  218 

V.  Fig.  219  shows  the  case  of  a  beam  supported  at  the  ends, 

and  loaded  with  three  concentrated 
loads  at  the  points  B,  C,  and  D  re- 
spectively; the  lines  of  bending-mo- 
ments for  each  individual  load  being 
respectively  AFE,  AGE,  and  AHE, 
FIG.  219.  and  the  actual  line  of  bending-mo- 

ments being  AKLME. 


REPRESENTATION  OF  BEND  ING-MOMENTS. 


289 


VI.    Fig.  220  shows  the  case  of  a  beam  supported  at  the 
ends,  and  loaded  with  a  uniformly  dis-      A  E          F       B 

tributed    load ;    the    line    of    bending- 
moments    being   a    curve,    ACDB,    as 


shown  in  the  figure.  FlG.  220> 


VII.  In  Fig.  221  we  have  the  case  of  a  beam,  over  a  part  of 
which,  viz.,  EF,  there  is  a  distributed  load ;    the  rest  of   the 

beam  being  unloaded.  The  line  of 
bending-moments  is  curvilinear  be- 
tween E  and  F,  and  straight  outside 


Xjs^/tt    y^      of  these  limits.     It  isAGSHB;  and, 
when  the  curve  is  plotted,  we  can 


N/ 

find   the   greatest   bending-moment 

graphically  by  finding  its  greatest  ordinate.  We  can  also 
determine  it  analytically  by  first  determining  the  bending- 
moment  at  a  distance  x  from  the  origin,  and  on  the  side 
towards  the  resultant  of  the  load,  and  then  differentiating. 
This  process  is  shown  in  the  following:  — 
Let  A  (Fig.  222)  be  the  point  where 
the  resultant  of  the  load  acts,  and  O  the 


middle  of    the   beam,   and    let   w  be    the  c  OA     B 

load  per  unit  of  length  ;  let  OA  =  a,  AB  = 

AC  =  b,  and  ED  =  2^,  so  that  the  whole  load  =  2wb :  there- 

a  A-  c       wb(a  4-  c] 
fore  supporting  force  at  D  =  2ivb  =  - -. 

If  we  take  a  section  at  a  distance  x  from  O  to  the  right,  we 
shall  have,  for  the  bending-moment  at  that  section, 

wb(a  -+•  c)  w 

(c  —  x) (a  -f  b  —  x)2  =  a  maximum. 

Differentiate,  and  we  have 

—  wb(a -\- c\  a(c  —  &) 

I         tioi  (  /T         j       h  */\  /~\  •  *     — 

c  T     V    -r          ;  ...        -          c 


290 


APPLIED  MECHANICS. 


hence  the  greatest  bending-moment  will  be 

^  \  C         )  2  \  C  ) 


7  (a 


—  ac 


— 


VIII.    In  Figs.  223  and  224  we  have  the  case  of  a  beam 
supported    at 
the  ends,  and 

1          J      J          •<.!_ 

loaded  with  a 
uniformly  dis- 
tributed load, 
and  also  with 
a  c  o  n  c  e  n- 
trated  load. 
In  the  first 


FiG.224. 


FlG-223- 


figure,  the  greatest  bending-moment 


is  at/?,  and  in  the  second  at  C. 


IX.  In  Fig.  225  we  have  a  beam  supported  at  A  and  B,  and 
loaded  at  C  and  D  with  equal 
weights;  the  lengths  of  AC  and 
BD  being  equal.  We  have,  con- 
sequently, between  A  and  £,  a 
uniform  bending-moment  ;  while 
on  the  left  of  A  and  on  the  right 


The  line  of  bending- 


FlG-  225' 

of  B  we  have  a  varying  bending-moment. 
moments  is,  in  this  case,  CabD. 

We  may,  in  a  similar  way,  derive  curves  of  bending-moment 
for  all  cases  of  loading  and  supporting  beams. 


AT  DIFFERENT   PARTS    OF  A    BEAM.        2QT 


§  190.  Mode  of  Procedure  for  Ascertaining  the  Stresses^ 
at  Different  Parts  of  a  Beam  when  the  Loads  and  the  Di- 
mensions are  given,  and  when  no  Fibre  at  the  Cross- 
section  under  Consideration  is  Strained  beyond  the 
Elastic  Limit.  —  When  the  dimensions  of  a  beam,  the 
load  and  its  distribution,  and  the  manner  of  supporting  are 
given,  and  it  is  desired  to  find  the  actual  intensity  of  the  stress 
on  any  particular  fibre  at  any  given  cross-section,  we  must  pro- 
ceed as  follows  :-- 

i°.  Find  the  actual  bending-moment  (M)  at  that  cross-sec- 
tion. 

2°.  Find  the  moment  of  inertia  (/)  of  the  section  about  its 
neutral  axis. 

3°.  Observe,  that,  from  what  has  already  been  shown,  the 
moment  of  the  couple  formed  by  the  tensions  and  compressions 
is  al,  where  a  =  intensity  of  stress  of  a  fibre  whose  distance 
from  the  neutral  axis  is  unity,  and  that  this  moment  must  equal 
the  bending-moment  at  the  section  in  order  to  secure  equilib- 
rium. Hence  we  must  have 


Moreover,  if  /  denote  the  (unknown)  intensity  of  the  stress 
of  the  fibre  where  the*  stress  is  desired,  and  if  y  denote  the 
distance  of  this  fibre  from  the  neutral  axis,  we  shall  have 


from  which  equation  we  can  determine/. 

EXAMPLES. 

i.  Given  a  beam  18  feet  span,  supported  at  both  ends,  and  loaded 
uniformly  (its  own  weight  included)  with  1000  Ibs.  per  foot  of  length. 
The  cross-  section  is  a  T,  where  area  of  flange  =  3  square  inches, 
area  of  web  =  4  square  inches,  height  =  10  inches.  Find  (a)  the 


292  APPLIED    MECHANICS. 


bending-moment  at  3  feet  from  one  end  ;  (b)  the  greatest  bending- 
moment;  (c)  the  greatest  intensity  of  the  tension  at  each  of  the 
above  sections  ;  (d)  the  greatest  intensity  of  the  compression  at  each 
>of  these  sections. 

2.  Given  an  I-beam  with  equal  flanges,  area  of  each  flange  =  3 
square  inches,  area  of  web  =  3  square  inches,  height  =  10  inches;  the 
beam  is  1 2  feet  long,  supported  at  the  ends,  and  loaded  uniformly  (its 
own  weight  included)  with  a  load  of  2000  Ibs.  per  foot  of  length.  Find 
J(a)  the  bending-moment  at  a  section  one  foot  from  the  end ;  (<£)  the 
greatest  bending-moment ;  (<r)  the  greatest  intensity  of  the  stress  at 
each  of  the  above  cross-sections. 


§  191.  Mode  of  Procedure  for  Ascertaining  the  Dimen- 
sions of  a  Beam  to  bear  a  Certain  Load,  or  the  Load  that 
a  Beam  of  Given  Dimensions  and  Material  is  Capable  of 
Bearing.  —  If  we  wish  to  determine  the*  proper  dimensions 
of  the  beam  when  the  load  and  its  distribution,  as  well  as  the 
manner  of  supporting,  are  given,  so  that  it  shall  nowhere  be 
strained  beyond  safe  limits,  or  if  we  wish  to  determine  the 
greatest  load  consistent  with  safety  when  the  other  quantities 
are  given,  we  must  impose  the  condition  that  the  greatest 
intensity  of  the  tension  to  which  any  fibre  is  subjected  shall 
not  exceed  the  safe  working-strength  for  tension  of  the  mate- 
rial of  which  the  beam  is  made,  and  the  greatest  intensity  of 
the  compression  to  which  any  fibre  is  subjected  shall  not  exceed 
the  safe  working-strength  of  the  material  (or  compression. 

Thus,  we  must  in  this  case  first  determine  where  is  the 
section  of  greatest  bending-moment  (this  determination  some- 
limes  involves  the  use  of  the  Differential  Calculus). 

Next  we  must  determine  the  magnitude  of  the  greatest 
bending-moment,  absolutely  if  the  load  and  length  of  the  beam 
are  given  (if  not,  in  terms  of  these  quantities),  and  then  equate 
this  to  the  moment  of  the  resisting  couple. 

Thus,  if  MQ  is  the  greatest  bending-moment,  when  the  loads 
are  such  that  no  fibre  is  strained  beyond  the  elastic  limit,  70  the 


WORKING-STRENGTH.  .      2$$ 

moment  of  inertia  of  that  section  where  this  greatest  bending-moment 
acts,  and  if  }t  =  greatest  tensile  fibre  stress  per  square  inch,  fc  = 
greatest  compressive  fibre  strength  per  square  inch,  yt  =  distance 
of  most  stretched  fibre  from  the  neutral  axis,  and  yc  =  distance 

of  most  compressed  fibre  from  the  neutral  axis,  then  will  —  be 

yt 

the  greatest  tension  per  square  inch,  at  a  unit's  distance  from  the 
neutral  axis,  and  —  the  greatest  compression  per  square  inch,  at  a 

unit's  distance  from  the  neutral  axis. 

Moreover,  in  this  case,  these  two  ratios  are  equal,  and  hence 

l/f         f*  T       ?c  T 
MQ  =  —I  =  —I. 

yt     yc 

SAFE  OR  WORKING-LOAD. 

If  //  =  safe  working-strength  per  square  inch  for  tension, 
/c'=safe  working-strength  per  square  inch  for  compression,  and 
M Q  =  greatest  safe  working  bending-moment,  then  the  ratios, 

—  and  — ,  are  not  equal. 

yt       yc 

f r  f '  f ' 

Hence,  when  —   is  less  than  —  we  have  M o'  = — /,  and  when. 

yt  yc  yt 

It'  ]c  fc 

—  is  greater  than  —  we  have  MQ  =—I. 

yt  yc  yc 

BREAKING-LOAD   AND   MODULUS   OF   RUPTURE. 

If  M  is  the  greatest  bending-moment  when  the  beam  is 
subjected  to  its  breaking-load,  the  formulae  given  above  do  not 
apply,  inasmuch  as  a  portion  of  the  fibres  are  strained  beyond 
the  elastic  limit,  and  Hooke's  law  no  longer  holds,  since,  after 
the  elastic  limit  is  passed,  the  ratio  of  stress  to  strain  decreases 
when  the  stress  increases. 

Indeed,  the  stresses  in  the  different  fibres  are  no  longer  pro- 


294  APPLIED    MECHANICS. 

portional  to  the  distances  of  those  fibres  from  the  neutral  axis. 
A  graphical  representation  of  the  stress  at  different  points  of  any 
given  section  AB  would  be  of  the  character  shown  in  the  figure, 
o  ,  the  form  of  the  curve  CDE  varying  with  the  shape 
of  the  cross-section. 

•    Nevertheless,  it  is  customary  to  compute  the 
breaking-strength   of    a   beam   by   means   of  the 

My 
formula  /=—  =—  ,  where  y  is  taken  as  the  distance  from  the  neutral 

axis  to  that  outer  fibre  which  gives  way  first,  i.e.,  to  the  most 
stretched  fibre  if  the  beam  breaks  by  tension,  or  to  the  most  com- 
pressed fibre,  if  it  breaks  by  compression.  The  quantity  /,  which 
may  thus  be  computed  from  the  formula 

My 


is  defined  as  the  Modulus  oj  Rupture. 

Inasmuch  as  this  formula  would  give  the  outside  fibre  stress, 
if  the  stress  were  uniformly  varying,  it  follows  that,  in  the  case  of 
materials  for  which  the  tensile  is  less  than  the  compressive  strength, 
the  modulus  of  rupture  is  greater  than  the  tensile  strength,  while 
in  that  of  materials  for  which  the  compressive  is  less  than 
the  tensile  strength  the  modulus  of  rupture  is  greater  than  the 
compressive  strength. 

For  experimental  work  bearing  upon  this  matter,  see  an 
article  by  Prof.  J.  Sondericker,  in  the  Technology  Quarterly  for 
October,  1888. 

WORKING-STRENGTH. 

The  working-strength  per  square  inch  of  a  material  for  trans- 
verse strength  is  the  greatest  stress  per  square  inch  to  which  it 
is  safe  to  subject  the  most  strained  fibre  of  the  beam.  It  is  usually 
obtained  by  dividing  the  modulus  of  rupture  by  some  factor  of 
safety,  as  3  or  4. 


WORKING-STRENGTH.  2$$ 

§   192.  EXAMPLES. 

i.  Given  a  beam  (Fig.  226)  supported  at  both  ends,  and  loaded, 
i°,  with  w  pounds  per  unit  of  length  uniformly,  and  2°,  with  a  single 
load  Wat  a.  distance  a  from  the  left-hand  support:  find  the  position 
of  the  section  of  greatest  bending-moment,  and  the  value  of  the  greatest 
bending-moment. 
o  A  a  Solution. 


1 


(i)  Left-hand  supporti ng- force  =  —  -  -\ — . 

Right-hand  supporting-force  =  _  _j_        t 


(2)  Assume  a  section  at  a  distance  x  from  the  left-hand  support 
(this  support  being  the  origin),  and  the  bending-moment  at  that  sec- 
tion is,  — 

(wl       W(l  -  a)\          wx* 
when  x  <  a,          —-\ 


2  £  )  2 

and  when  x  >  a, 

wx* 


To  find  the  value  of  x  for  the  section  of  greatest  bending-moment, 
differentiate  each,  and  put  the  first  differential  co-efficient  =  zero. 
We  shall  thus  have,  in  the  first  case, 

Wl       W(l  -a)  I       W(l  -  a) 

--  1  --  ~  -  -  —  w#  =  o,  or  x  =  -  H  --  -  —  -.  —  -  : 
2  /  2  wl 

and  in  the  second  case, 

wl       W(l  -a)  I       W(l  -a)        W 

-  +  —  —,  -  -  -wx-  W**  o,  or  x  =  -  +       v     .  —  -  --  . 

2  /  2  Wl  W 

Now,  whenever  the  first  is  <  #,  or  the  second  is  >  a,  we  shall  have 
in  that  one  the  value  of  x  corresponding  to  the  section  of  greatest 
bending-moment.  But  if  the  first  is  >  a,  and  the  second  <  0,  then  the 
greatest  bending-moment  is  at  the  concentrated  load. 

These  conclusions  will  be  evident  on  drawing  a  diagram  representing 
the  bending-moments  graphically,  as  in  Figs.  223  and  224;  and  the 
greatest  bending-moment  may  then  be  found  by  substituting,  in  the  cor- 
responding expression  for  the  bending-moment,  the  deduced  value  of  x. 


296  APPLIED    MECHANICS. 

2.   Given  an  I-beam,  10  feet  long,  supported  at  both  ends,  and 
loaded,  at  a  distance  2  feet  to  the  left  of  the  middle,  with  20000  pounds. 
Find  the  bending-moment  at  the  middle,  the  greatest  bending-moment, 
also  the  greatest  intensity  of  the  tension,  and  that  of  the  compression  at 
each  of  these  sections. 

Given  Area  of  upper  flange  =  8  sq.  in. 

Area  of  lower  flange  =  5  sq.  in. 

Area  of  web  =  7  sq.  in. 
Total  depth  =  14  in. 

§  193.  Beams  of  Uniform  Strength.  —  Abeam  of  uniform 
strength  (technically  so  called)  is  one  in  which  the  dimensions 
of  the  cross-section  are  varied  in  such  a  manner,  that,  at  each 
cross-section,  the  greatest  intensity  of  the  tension  shall  be 
the  same,  and  so  also  the  greatest  intensity  of  the  com- 
pression. 

-Such  beams  are  very  rarely  used  ;  and,  as  the  cross-section 
varies  at  different  points,  it  would  be  decidedly  bad  engineering 
to  make  them  of  wood,  for  it  would  be  necessary  to  cut  the 
wood  across  the  grain,  and  this  would  develop  a  tendency  to 
split. 

In  making  them  of  iron,  also,  the  saving  of  iron  would  gen- 
erally be  more  than  offset  by  the  extra  cost  of  rolling  such  a 
beam.  Nevertheless,  we  will  discuss  the  form  of  such  beams  in 
the  case  wlien  the  section  is  rectangular. 

In  all  cases  we  have  the  general  equation 


y 

applying  at  each  cross-section,  where  M  =  bending-moment 
''section  at  distance  x  from  origin),  /  =  moment  of  inertia  of 
same  section,  '  y  =  distance  from  neutral  axis  to  most  strained 
fibre,  and  p  —  intensity  of  stress  on  most  strained  fibre  ;  the 
condition  for  this  case  being  that  /  is  a  constant  for  all  values 
of  x  (i.e.,  for  all  positions  of  the  section),  while  M,  /,  and  y 
are  functions  of  x. 


BEAMS   OF  UNIFORM  STRENGTH.  297 

As  we  are  limiting  ourselves  to  rectangular  sections,  if  we 
let  b  =  breadth  and  h  =  depth  of  rectangle  (one  or  both  vary- 
ing with  x\  we  shall  have 


as  the  condition  for  such  a  beam,  with/  a  constant  for  all  values 
of  x,  when  the  same  load  remains  on  the  beam. 

We  must,  therefore,  have  bk2  proportional  to  M.     Hence, 
assuming  the  origin  as  before, 

i°.  Fixed  at  one  end,  load  at  the  other,          bh*  =(-)  W(l  —  *). 


2°.  Fixed  at  one  end,  uniformly  loaded,          bh2  =  (  -  —  )  (/  —  x)2. 

\        21' 


, 

3°.  Supported  at  ends,  loaded  at  I  2  \/  2 

middle-  | 

2  p     2 

4°.  Supported  at  ends,  uniformly  loaded,       bh2  =  (  ---  }(lx  —  x2). 

\j>  2l  ' 

Now,  this  variation  of  section  may  be  accomplished  in  one 
of  two  ways:  ist,  by  making  h  constant,  and  letting  b  vary; 
and  2d,  by  making  b  constant,  and  letting  h  vary.  Thus,  in 
the  first  case  above  mentioned,  if  h  is  constant,  we  have,  for  the 
plan  of  the  beam, 


and  if  one  side  be  taken  parallel  to  the  axis  of  the  beam,  this 
will  be  the  equation  of  the  other  side ;  and,  as  this  is  the  equa- 
tion of  a  straight  line,  the  plan  will  be  a  triangle. 


APPLIED  MECHANICS. 


If,  on  the  other  hand,  b  be  constant,  and  h  vary,  we  shall 
nave,  for  the  vertical  longitudinal  section  of  the  beam, 


and,  if  one  side  be  taken  as  a  straight  line  in  the  direction  of 
the  axis,  the  other  will  be  a  parabola. 

A  similar  reasoning  will  give  the  plan  or  elevation  respect- 
ively in  each  case ;  and  these  can  be  readily  plotted  from  their 
equations. 

CROSS-SECTION  OF  EQUAL  STRENGTH. 

A  cross-section  of  equal  strength  (technically  so  called)  is 

one  so  proportioned  that  the  greatest  intensity  of  the  tension 

shall  bear  the  same  ratio  to  the  breaking  tensile  strength  of  the 

material  as  the  greatest  intensity  of  the  compression  bears  to 

the  breaking  compressive  strength  of   the  material.     This  is 

accomplished,  as  will  be  shown  directly,  by  so  arranging  the 

form  and  dimensions  of  the  section  that  the  distance  of  the 

neutral  axis  from  the  most  stretched  fibre  shall  bear  to    its 

distance  from  the  most  compressed  fibre  the  same  ratio  that 

the  tensile  bears  to  the  compressive  strength  of  the  material. 

Let  fc  —  breaking-strength  per  square  inch  for  compression, 

ft  =  breaking-strength  per  square  inch  for  tension, 

yc  =  distance  of   neutral  axis  from  most   compressed 

fibre, 

yt  =  distance  of  neutral  axis  from  most  stretched  fibre. 

If  pc  =  actual  greatest  intensity  of  compression,  and  pt  = 

actual  greatest  intensity  of   tension,  then,  for  a  cross-section 

of  equal  strength,  we  must  have,  according  to  the  definition, 

<£=•£;  but  we  have  —  =  —  =  intensity  of  stress  at  a  unit's 

pt    ft  yc     yt 


CROSS-SECTION  OF  EQUAL   STRENGTH.  2Q9 

distance  from  the  neutral  axis.     Hence,  combining  these  two, 
we  obtain 

y-  =  7 

yt     ft 

EXAMPLE. 

Suppose  we  have^  =  80000  Ibs.  per  square  inch,  and/  =  20000 
Ibs.  per  square  inch. :  find  the  proper  proportion  between  the  flange  At 
and  the  web  A2  of  a  T-section  whose  depth  is  h. 

§  194.  Deflection  of  Beams.  — We  have  already  seen  (§  185), 
that,  in  the  case  of  a  beam  which  is  bent  by  a  transverse  load, 
we  have 

-'oo, 


a 

r 


where  (having  assumed  a  certain  cross-section  whose  distance 
from  the  origin  is  x)  a  =  the  strain  of  a  fibre  whose  distance 
from  the  neutral  axis  is  y,  and  r  =  radius  of  curvature  of 
the  neutral  lamina  at  the  section  in  question.  Hence  follows  the 
equation 


but  from  the  definition  of  Et  the  modulus  of  elasticity,  we  shall 
have 

V* 

where  /  =  intensity  of  the  stress, at  a  distance  y  from  the 
neutral  axis. 

Hence  it  follows,  assuming  Hooke's  law,  that 

r       Ey       E  y 
We  have  already  seen,  that,  disregarding  signs,  M  =  -  / 


3^0  APPLIED   MECHANICS. 

(making,  of  course,  the  two  assumptions  already  spoken  of 
when  this  formula  was  deduced),  where  M  =  bending-moment 
at,  and  /  =  moment  of  inertia  of,  the  section  in  question  ;  i.e., 
of  that  section  whose  distance  from  the  origin  is  x.  This  gives 

-  = ,  if,  denoting  tension  by  the  +  sign>  and  taking  y 

positive  upwards,  we  call  M  positive  when  it  tends  to  cause 
tension  on  the  lower,  and  compression  on  the  upper,  side;  these 
being  the  conventions  in  regard  to  signs  which  we  shall  adopt 
in  future.  Hence,  by  substitution,  we  have 

1  -  p  •      -  M  (i\ 

~r~Ey-        El 

Now,  if  we  assume  the  axis  of  x  coincident  with  the  neutral 
line  of  the  central  longitudinal  section  of  the  beam,  and  the 
axis  of  v  at  right  angles  to  this,  and  v  positive  upwards,  no 
matter  where  the  origin  is  taken,  we  shall  always  have,  as  is 
shown  in  the  Differential  Calculus, 


'  (•+(1)7 

Hence  equation  (i)  becomes 


(*) 


M  and  /  being  functions  of  x :  and,  when  we  can  integrate 
this  equation,  we  can  obtain  v  in  terms  of  x,  thus  having  the 
equation  of  the  elastic  curve  of  the  neutral  line ;  and,  by  com- 
puting the  value  of  v  corresponding  to  any  assumed  value  of  x, 
we  can  obtain  the  deflection  at  that  point  of  the  beam. 


FORMULA   FOR  SLOPE   AND   DEFLECTION.  3OI 

The  above  equation  (2)  is,  as  a  rule,  too  complicated  to  be 
integrated,  except  by  approximation  ;  and  the  approximation 
usually  made  is  the  following  :  — 

Since  in  a  beam  not  too  heavily  loaded,  the  slope,  and  con- 
sequently the  tangent  of  the  slope  (or  angle  the  neutral  line 
makes  with  the  horizontal  at  any  point),  is  necessarily  small,  it 

follows  that  —  is  very  small,  and  hence  (--)  is  also  very  small, 
dx  \dxl 

and  i  +  (  —  )   is  nearly  equal  to  unity.     Making  this  substitu- 
tion, we  obtain,  in  place  of  equation  (2), 


-. 

d*  ~  EI* 

and  this  is  the  equation  with  which  we  always  start  in  com- 
puting the  slope  and  deflection  of  a  loaded  beam,  or  in  finding 
the  equation  of  the  elastic  line. 

By  one  integration  (suitably  determining  the  arbitrary  con- 

stant) we  obtain  the  slope  whose  tangent  is  —,  and  by  a  second 

dx 

integration  we  obtain  the  deflection  v  at  a  distance  x  from  the 
origin  ;  and  thus,  by  substituting  any  desired  value  for  x,  we 
can  obtain  the  deflection  at  any  point. 

§  195.  Ordinary    Formulae   for    Slope   and    Deflection.  — 
We  may  therefore  write,  if  i  is  the  circular  measure  of  the 

slope  at  a  distance  x  from  the  origin,  since  i  =  tan  i  =  -j- 

dx 

nearly, 

<£v  =  M_ 

dx2  ~~~  Ef 


f 


M 


3O2  APPLIED   MECHANICS. 

In  these  equations,  of  course,  E  is  taken  as  a  constant,  M 
must  ALWAYS  be  expressed  in  terms  of  x,  and  so  also  must  / 
whenever  the  section  varies  at  different  points.  When,  how- 
ever, the  section  is  uniform,  /  is  constant,  and  the  formulae 
reduce  to 


"Jiff- 


§  196.  Special  Cases  --  i°.  Let  us  take  a  cantilever  loaded 
with  a  single  load  at  the  free  end.  Assume  the  origin,  as 
before,  at  the  fixed  end,  and  let  the  beam  be  one  of  uniform 
section.  We  then  have  M  =  —  W(l  —  x\ 


*'~<>  -*<—    ?- 

To  determine  c,  observe  that  when  x  =  o,  i  =  o  ; 


c  •=•  o 


is  the  slope  at  a  distance  x  from  the  origin. 
The  deflection  at  the  same  point  will  be 

, .  f,**  =  — £  0  -  £y&  =  -^-2  -  ^w  „ 

J  EIJ  \          2  )  EI\  2         6  ) 

but  when  x  =  o,  v  =  o        .*.    ^  =  o        /.     the  deflection  at 
a  distance  x  from  the  origin  will  be 


The  equations  (i)  and  (2)  give  us  the  means  of  finding  the 
slope  and  deflection  at  any  point  of  the  beam. 

To  find  the  greatest  slope  and  deflection,  we  have  that  both 
expressions  are  greatest  when  x  =  I.  Hence,  if  *0  and  VQ  rep- 
resent the  greatest  slope  and  deflection  respectively, 

Wl2 


SPECIAL    CASES.  3°3 


2°.  Next  take  the  case  of  a  beam  supported  at  both  ends 
and  loaded  uniformly,  the  load  per  unit  of  length  being  w. 
Assume  the  origin  at  the  left-hand  end  ;  then 


wl        wx2       w 
M  —  —x  ---  =  —  Ux  —  x2)          and         W=  wl 

2  2  2  V 


w 


/w    /lx2       x*\ 
(lx-x^x  =  _(___)  +  ,. 


2EI  ^ 

To  determine  c,  we  have  that  when  x  =  -,  then  i  =  o; 

W      //3  /3\  2£//3 

V  8  "  V  +  £ 
-  — }  -      W^  W     (6/x2  -  /3^ 

f  ff '/ '  V    -5  "?/  r>  A  7?  T  <y  A  Ji '  7 ^  ^  '        V    / 


/w      /* 
to  =  ^£7j  (6^  "  ^  ~  /3)^ 


24^7 

But  when  x  =  o,  v  =  o ; 


/.    r  =  o 


For  the  greatest  slope,  we  have  ;tr  =  o,  or  x  =  I; 


24^*7      24^7 


For  the  greatest  deflection,  x  =  -  ; 


— w  5/4       —5 a//4       — 

384^7 


304  APPLIED    MECHANICS. 

3°.  Take  the  case  of  a  beam  supported  at  both  ends,  and 
loaded  at  the  middle  with  a  load  W. 

Assume,  as  before,  the  origin  at  the  left-hand  support. 
Then  we  shall  have 

W  I  W  I 

•M  =  —x,      x  <  -,     and     M  —  —  (/  —  x}  when  x  >  — 

Therefore,  for  the  slope  up  to  the  middle,  we  have 

w  r          w  x2 

i  =  —^r,  I  xdx  =  —=-T h  c. 

2EIJ  2EI  2 

When  x  ~  — ,  then  i  =  o ; 

wr 


W . 


and 


w  /v  ,    r\.       w  (x>    rx\ 

v  —  —    -  I  \x lax  =  — —I 

AJcl'J    \         •  4/  4.fi/\3          4/ 

But  when  x  —  o,  v  =  o ; 


c  =  o. 


i        4 

The  slope  is  greatest  when  x  —  o ; 

.'.     z'    = 


The  deflection  is  greatest  when  x  =  -; 


4°.  In  the  following  table  7  denotes  the  moment  of  inertia 
of  the  largest  section  : 


SPECIAL    CASES. 


305 


Uniform  Cross-Section. 

Greatest  Slope. 

Greatest 
Deflection. 

Fixed  at  one  end,  loaded  at  the  other 
Fixed  at  one  end,  loaded  uniformly    .     . 
Supported  at  ends,  load  at  middle  .     .     . 
Supported  at  ends,  uniformly  loaded  .     . 

i  Wl2 
2  7?7 
i  Wl2 

i  Wl* 
ZEI 

i  Wl* 

*>EJ 
i  Wl2 

*EI 
i  Wl* 

'6  El 
i  Wl* 
^EI 

48^7 
5  #73 
384^/ 

Uniform  Strength  and   Uniform   Depth, 
Rectangular  Section. 

Fixed  at  one  end,  load  at  the  other     .     . 
Fixed  at  one  end,  uniformly  loaded    .     . 
Supported  at  both  ends,  load  at  middle  . 
Supported  at  both  ends,  uniformly  loaded, 

Wl2 
~EI 
i  Wl2 
2  El 
i  Wl* 
*~EI 
i  Wl2 

i  Wl* 
~*~EI 
iWZ* 

4^7 
i  Wl* 
V~EI 

i  m* 

&  EI 

64^7 

Uniform  Strength  and  Uniform  Breadth, 
Rectangular  Section. 

Fixed  at  one  end,  loaded  at  the  other, 
Supported  at  both  ends,  load  at  middle, 
Supported  at  both  ends,  uniformly  loaded 

wr 

2~EI 

i  wr 

2  wr 

3   EI 

i  wr 

4  El 

o  wr 

24  EI 
0018  Wr 

0-098  EI 

0.010    EJ 

306  APPLIED   MECHANICS. 

§197.    Deflection  with   Uniform   Bending-Moment If 

the  bending-moment  is  uniform,  then  M  is  constant ;  and,  if  / 
is  also  constant,  we  have 

•_  ^L  f       -  Mx 

but  when  x  =  -,  then  i  =  o; 

Ml 


2EI 


Ml        l\       dv 

•"•     t  =  T^rl  x  --  )  =  -r 
EI\         2]       dx 


lx 


the  constant  disappearing  because  v  =  o  when  x  =  o. 

Hence,  for  a  beam  where  the  bending-moment  is  uniform, 
we  have 

•  _  ¥J        ^\  M Y*2      ^ 

and  for  greatest  slope  and  deflection,  we  have 

-Ml  Mil*       /2\  i  Ml2 

1       —     .71      —    

^O     * ~~  r\  T  *  V<~\     •—        -r- 


§  198.  Resilience  of  a  Beam.  —  7^  resilience  of  a  beam 
is  the  mechanical  work  performed  in  deflecting  it  to  the  amount 
it  would  deflect  under  its  greatest  allowable  gradually  applied 
load.  In  the  case  of  a  concentrated  load,  if  W  is  the  greatest 
allowable  gradually  applied  load,  and  vl  the  corresponding 
deflection  at  the  point  of  application  of  the  load,  then  will  the 

W 

mean  value  of  the  load  that  produces  this  deflection  be  — » 

W 

and  the  resilience  of  the  beam  will  be  — z/,. 

2 


SLOPE  AND   DEFLECTION  OF  A   BEAM.  3O/ 


§  i99-   Slope    and    Deflection    of   a    Beam  with   a   Con- 
centrated   Load    not    at    the 

Middle.  —  Take,   as    the   next      °  A a 

case,  a  beam  (Fig.   228).     Let        <  a 

the  load  at  A  be  W,  and  dis- 
tance OA  =  ay  and  let  a  >  -. 

2  FIG.  228. 

W(l  -  a) 
x  <  a    M  =  — ^ '-  x, 


^f 


-  a) 

••- x<a  '= 


When  x  =  o,     *  =  4  =  undetermined  slope  at 

=  •  •      .  =  W(l  -  a 

and 


When  x  =.  o,     v  =  o  ; 


To  determine  r,  observe  that  when  ^r  —  ^,  this  value  of  i 
and  that  deduced  from  (i)  must  be  identical. 

Waf     _^!\,  W(l  -  a)a*        .  Wa*        . 

TEI\ a      2  )  H  2/^7  2^7 


308  APPLIED   MECHANICS. 


Wat ,          x*\         Wa*    ,     .     , 


2    /  2EI 

or 


and 

v  =  Jlx  -**-  ld)dx 


To  determine  c,  observe  that  when  x  =  a,  this  value  of  v 
and  (2)  must  be  identical ; 

Wa  f       «3\        .  W(l  —  a) 


/       a 

( 

\       i 


—  «4  4-  tf4)  = 


61EI       6EI 
+  ^2)  H-  ^-          (4) 
To  determine  4>  we  have  that  when  x  —  /,  v  =  o  ; 


Substituting  this  value  of  i0  in  the  equations  (i),  (2),  (3),  and 
(4),  we  obtain  for 


SLOPE   AND   DEFLECTION  OF  A    BEAM.  309 


Wa  ,    r  Wa 


(4)   v  = 


To  find  the  greatest  deflection,  differentiate  (2),  and  place 
the  first  differential  co-efficient  equal  to  zero  :  or,  which  is  the 
same  thing,  place  i  =  o  in  (i),  and  find  the  value  of  x  ;  then 
substitute  this  value  in  (2),  and  we  shall  have  the  greatest 
deflection. 

We  thus  obtain 

(/-*)*>  =  ^(3«/  -*/•-*>)        .-.    «•  =  "-(*  ~  &  +  a-\ 
3  3\        '  —  '         / 

or 


•'•     *=' 'a~'> 


and  the  greatest  deflection  becomes 

Wa(l  -  a)(2l  -  a) 


_ 


§  2OO.  EXAMPLES. 

1.  In  example  i,  p.  294,  find  the  greatest  deflection  of  the  beam 
when  it  is  loaded  with  \  of  its  breaking-load,  assuming  E  =  1200000. 

2.  In  the  same  case,  find  what  load  will  cause  it  to  deflect  ^J^  of  its 
span. 

3.  What  will  be  the  stress  at  the  most  strained  fibre  when  this  occurs- 

4.  In  example  3,  p.  294,  find  the  load  the  beam  will  bear  without 
deflecting  more  than  ^J^  of  its  span,  assuming  E  =  24000000. 

5.  Find  the  stress  at  the  most  strained  fibre  when  this  occurs. 

6.  In  example  6,  p.  295,  find  the  greatest  deflection  under  a  load 
J  the  breaking-load. 


3io 


APPLIED   MECHANICS. 


§  2Oi.   Deflection   and    Slope   under  Working-Load.  —  If 

we  take  the  four  cases  of  deflection  given  in  the  first  part  of 

the  table  on  p.  305,  and  calling/  the  working  strength  of  the 

material,  and  y  the  distance  of  the  most  strained  fibre  from 
the  neutral  axis,  and  if  we  make  the  applied  load  the  working- 
load,  we  shall  have  respectively  — 

!°    m=*L         •  W=^ 

y  ty 

Wl      //  2/7 

2   .    =  —  /.  W  —  — — 

2         y  ly 

m=fj 

4  '"  y 
wi     fl 


4-   T«-T 


*y 

:     W=  ^ 


And  the  values  of  slope  and  deflection  will  become  respectively, 


Slope. 


Deflection. 


Slope. 


Deflection. 


/» 


2. 


From  these  values,  and  those  given  on  p.  305,  we  derive  the 
following  two  propositions  :  — 

i°.  If  we  have  a  series  of  beams  differing  only  in  length; 
and  we  apply  the  same  load  in  the  same  manner  to  each,  their 
greatest  slopes  will  vary  as  the  squares  of  their  lengths,  and 
their  greatest  deflections  as  the  cubes  of  their  lengths. 


SLOPE   AND  DEFLECTION  OF  RECTANGULAR  BEAMS.   31! 


2°.  If,  however,  we  load  the  same  beams,  not  with  the  same 
load,  but  each  one  with  its  working-load,  as  determined  by 
allowing  a  given  greatest  fibre  stress,  then  will  their  greatest 
slopes  vary  as  the  lengths,  and  their  greatest  deflections  as  the 
squares  of  their  lengths. 

§  202.   Slope    and    Deflection   of   Rectangular   Beams 

bfc  h 

If  the  beams  are  rectangular,  so  that  /  =  —  and  y  =  -,  the 

values  of  slope  and  deflection  above  referred  to  become  further 
simplified,  and  we  have  the  following  tables  :  — 


Given  Load  W. 

Working-Load. 
Greatest  Fibre  Stress  =/.  * 

Slope. 

Deflection. 

Slope. 

Deflection. 

1°. 

2°. 

3°. 
4°- 

6W12 

4  #73 

I 

2/2 

Ebfc 
2W* 

Ebte 

3  Wl* 

Eh 

2_fl 

3  Eh 

i£ 

Ebte 

3  Wl* 
4£6fc 
i   Wl* 

*Ebh* 
i  Wl* 

3  Eh 
i  fl 

2  Eh 
i/' 

lEbfc 

5  wr* 

*Eh 

2ft 

1  Eh 

(>Eh 

sfi* 

^Eh 

*  Ebfc 

32  Ebh* 

So  that,  in  the  case  of  rectangular  beams  similarly  loaded  and 
supported,  we  may  say  that  — 

Under  a  given  load  W,  the  slopes  vary  as  the  squares  of 
the  lengths,  and  inversely  as  the  breadths  and  the  cubes  of  the 
depths ;  while  the  deflections  vary  as  the  cubes  of  the  lengths, 
and  inversely  as  the  breadths  and  the  cubes  of  the  depths. 


312  APPLIED  MECHANICS. 

On  the  other  hand,  under  their  working-loads,  the  slopes  vary 
directly  as  the  lengths,  and  inversely  as  the  depths  ;  while  the 
deflections  vary  as  the  squares  of  the  lengths,  and  inversely  as 
the  depths. 

§  203.  Beams  Fixed  at  the  Ends.  —  The  only  cases  which 
we  shall  discuss  here  are  the  two  following ;  viz.,  — 

i°.  Uniform  section  loaded  at  the  middle. 

2°.   Uniform  section,  load  uniformly  distributed. 

CASE  I.  —  Uniform  Section  loaded  at  the  Middle.  —  The 
fixing  at  the  ends  may  be  effected  by  building  the  beam  for 

some  distance  into  the  wall,  as 
shown  in  Fig.  229.     The  same 
result,  as  far  as  the  effect  on 
|w  the  beam  is  concerned,  might 

be   effected    as   follows :   Hav- 
ing merely  supported    it,   and 

placed  upon  it  the  loads  it  has  to  bear,  load  the  ends  outside 
of  the  supports  just  enough  to  make  the  tangents  at  the  sup- 
ports horizontal. 

These  loads  on  the  ends  would,  if  the  other  load  was  re- 
moved, cause  the  beam  to  be  convex  upwards :  and,  moreover, 
the  bending-moment  due  to  this  load  would  be  of  the  same 
amount  at  all  points  between  the  supports ;  i.e.,  a  uniform 
bending-moment.  Moreover,  since  the  effect  of  the  central 
load  and  the  loads  on  the  ends  is  to  make  the  tangents  over 
the  supports  horizontal,  it  follows  that  the  upward  slope  at  the 
support  due  to  the  uniform  bending-moment  above  described 
must  be  just  equal  in  amount  to  the  downward  slope  due  to  the 
load  at  the  middle,  which  occurs  when  the  beam  is  only  sup- 
ported. 

Hence  the  proper  method  of  proceeding  is  as  follows  :  — 

i°.  Calculate  the  slope  at  the  support  as  though  the  beam 
were  supported,  and  not  fixed,  at  the  ends ;  and  we  shall 
if  we  represent  this  slope  by  iu  the  equation 


BEAMS  FIXED   AT   THE  ENDS.  313 


w* 


2°.  Determine  the  uniform  bending-moment  which  would 
produce  this  slope. 

To  do  this,  we  have,  if  we  represent  this  uniform  bending- 
moment  by  Mlf  that  the  slope  which  it  would  produce  would  be 


and,  since  this  is  equal  to  *„  we  shall  have  the  equation 

_^/_  ;w_.  M 

TEI     -*«"-°  V3) 


.:    Jf,  =  ~  (4) 

This  is  the  actual  bending-moment  at  either  fixed  end  ;  and  the 
bending-moment  at  any  special  section  at  a  distance  x  from 
the  origin  will  be 


where  M  is  the  bending-moment  we  should  have  at  that  sec- 
tion if  the  beam  were  merely  supported,  and  not  fixed.  Hence, 
when  it  is  fixed  at  the  ends,  we  shall  have,  for  the  bending- 
moment  at  a  distance  x  from  O,  where  O  is  at  the  left-hand 
support, 

W        W, 
M=—oc-—l.  (5) 

When  x  =  -,  we  obtain,  as  bending-moment  at  the  middle, 

*-•?;  (6) 

o 

and,  since  Ml  =  —M0,  it  follows  that  the  greatest  bending- 
moment  is 

W 

8; 


3H  APPLIED   MECHANICS. 

this  being  the  magnitude  of  the  bending-moment  at  the  middle 
and  also  at  the  support. 

POINTS    OF    INFLECTION. 

The  value  of  M  becomes  zero  when 

x  =   -  and  when  x  =  — ; 
4  4' 

hence  it  follows  that  at  these  points  the  beam  is  not  bent,  and 
that  we  thus  have  two  points  of  inflection  half-way  between  the 
middle  and  the  supports. 

SLOPE    AND    DEFLECTION    UNDER    A    GIVEN    LOAD. 

We  shall  have,  as  before, 

W&       Wlx  . 


/M   , 
EIdX  = 


and  since,  when  x  =  o,  i  =  o, 
.«.    c '—  o 

"""     *  =  ~dx  = 


W  I  2Xl  IX2 

v  —  I 


3 


2 


the  constant  vanishing  because  v  =  o  when  x  =  o.     The  slope 

becomes  greatest  when  x  =  -,  and  the  deflection  when  x  =  -. 

4 
Hence  for  greatest  slope  and  deflection,  we  have 

Wl2  f  . 

64^7' 


BEAMS  FIXED   AT  THE  ENDS.  315 


SLOPE    AND    DEFLECTION    UNDER    THE    WORKING-LOAD. 

If  f  represent   the  working-strength  of   the   material   per 
square  inch,  and  if  W  represent  the  centre  working-load,  we 

shall  have 

fiP7=/7 

8    "  y 


CASE  II.  —  Uniform  Section,  Load  uniformly  Distributed.  — 
Pursuing  a  method  entirely  similar  to  that  adopted  in  the  former 
case,  we  have  — 

i°.  Slope  at  end,  on  the  supposition  of  supported  ends,  is 

Wl* 


24^/ 
2°.  Slope  at  end  under  uniform  bending-moment  Mt  is 

(») 


Hence,  since  their  sum  equals  zero, 

Wl 

12 ' 

which  is  the  bending-moment  over  either  support. 
The  bending-moment  at  distance  x  from  one  end  is 

W                       Wl 
M  —  — -  \lx       x2)  —  .  ^4) 

2/  12 

Wl 
This  is  greatest  when  x  =  o,  and  is  then .     Hence  great- 

1 2 

est  bending-moment  is,  in  magnitude, 

—  (5) 

12 


316  APPLIED   MECHANICS. 


POINTS    OF    INFLECTION. 

M  becomes  zero  when  x  =  -  ±  — =.  (6) 

2  2  ^3 

Hence  the  two  points  of  inflection  are  situated  at  a  distance 
/ 

on  either  side  of  the  middle. 


SLOPE    AND    DEFLECTION. 

,   M  ™7 

t  s=    I  -~dx  = 


the  constant  vanishing  because  z  =  o  when  x  =  Oi 
W 


v  = 


the  constant  vanishing  because  ^  =  o  when  .r  =  o.     Hence  for 
greatest  slope  and  deflection  we  have,  t  is  greatest  when  x  = 


-f  i  zt  -y=\ 


and  z;  is  greatest  when  *  =  -  ; 


SLOPE    AND    DEFLECTION    UNDER    WORKING-LOAD. 

For  working-load  we  have 

Wl       fl 

77  =  7 


B ENDING-MOMENT  AND   SHEARING-FORCE. 


3'7 


EXAMPLES. 

1.  Given  a  4-inch  by  12 -inch  yellow-pine  beam,  span  20  feet,  fixed 
at  the  ends ;  find  its  safe  centre  load,  its  safe  uniformly  distributed  load, 
and  its  deflection  under  each  load.     Assume  a  modulus  of  rupture  5000 
Ibs.  per  square  inch,  and  factor  of  safety  4.     Modulus   of  elasticity, 
1200000. 

2.  Find  the  depth  necessary  that  a  4-inch  wide  yellow-pine  beam,  20 
feet  span,  fixed  at  the  ends,  may  not  deflect  more  than  one  four-hun- 
dredth of  the  span  under  a  load  of  5000  Ibs.  centre  load. 

§  204.  Variation  of  Bending-Moment  with  Shearing- 
Force.  —  If,  in  any  loaded  beam  whatever,  M  represent  the 
b ending-moment,  and  F  the  shearing-force  at  a  distance  x  from 
the  origin,  then  will 

*-<*  «> 

Proof  (a).  —  In  the  case  of  a  cantilever  (Fig.  230),  assume 
the  origin  at  the  fixed  end ;  then,  if  M 
represent  the  bending-moment  at  a 
distance  x  from  the  origin,  and  M '+  ^M 
that  at  a  distance  x  +  t±x  from  the 
origin,  we  shall  have  the  following 
equations  :  — 

x  =  l 

M=  — S       W(a-  x), 

X  =  X 

x  =  l 

M +  AJ/  =  -2        W(a-  x  —  A#)  nearly. 

X  =  X 

a  being  the  co-ordinate  of  the  point  of  application  of  W, 

x  =  l 

AJ/=  AjeS        W  nearly 


====-  =  2       W: 


3  I  8  APPLIED   MECHANICS. 


and,  if  we  pass  to  the  limit,  and  observe  that 


we  shall  obtain 


(b)  In  the  case  of  a  beam  supported  at  the  ends  (Fig.  231), 
,    A.,*.  assume  the  origin   at  the  left-hand 

/|^ I ^j,j          7\   end,  and  let  the  left-hand  support- 
ing-force be  S ;  then,  if  a  represent 
FIG.  231.  the  distance  from  the  origin  to  the 

point  of  application  of   W,  we  shall  have  the  equations 


M  =  Sx  -  2        W(x  -  a), 

M  4-  bM  =  S(*  +  A#)  -  S       ^(tf  —  d5  4-  A*)  nearly. 
Hence,  by  subtraction, 

X-  X 

=  S .  tec  —  2       WA*  nearly 


JT  =  o 


=  o  —  2        ^nearly; 

mt-'  x  =  o 

and,  if  we  pass  to  the  limit,  and  observe  that 
p  ^  g 2       iff 

Jf  =  O 

we  shall  obtain 
as  before. 


LONGITUDINAL   SHEARING    OF  BEAMS.  319 

§  205.  Longitudinal  Shearing  of  Beams. — The  resistance 
of  a  beam  to  longitudinal  shearing  sometimes  becomes  a  mat- 
ter of  importance,  especially  in  timber,  where  the  resistance  to 
shearing  along  the  grain  is  very  small.  We  will  therefore  pro- 
ceed to  ascertain  how  to  compute  the  intensity  of  the  longi- 
tudinal shear  at  any  point  of  the  beam,  under  any  given  load  ; 
as  this  should  not  be  allowed  to  exceed  a  certain  safe  limit,  to 
be  determined  experimentally.  Assume  a  A 


section  AC  (Fig.  232)  at  a  distance  x  from        V 


the  origin,  and  let  the  bending-moment  at 
that  section  be  M.     Let  the  section  BD  be 

at  a  distance  x  +  &>*  from  the  origin,  and  

let  the  bending-moment  at  that  section  be  FIG.  232. 

M  +  kM. 

Let  y0  be  the  distance  of  the  outside  fibre  from  the  neutral 
axis ;  and  let  ca  —  y^  be  the  distance  of  a,  the  point  at  which 
the  shearing-force  is  required,  from  the  neutral  axis. 

Consider  the  forces  acting  on  the  portion  ABba,  and  we 
shall  have  — 


0 
i°.  Intensity  of  direct  stress  at  A  =  -j^. 

2°.  Intensity  of  direct  stress  at  a  unit's  distance  from  neu- 

M 
tral  axis  =  -j. 

My 
3°.  Intensity  of  direct  stress  at  ^,  where  ce  =  y,  is  —=-. 

(M  + 
So,  likewise,  intensity  of  direct  stress  at  /  is  —    — 

Therefore,  if  z  represent  the  width  of  the  beam  at  the  point 
?,  we  shall  have  — 

M  (*y° 

Total  stress  on  face  Aa  =  -j-  I    yzdy, 

1  Jyi 
M  +  ^M  r?o 

Total  stress  on  face  Bb  =  -  j  -  I    yzdy  ; 

l        Jyi 


320  APPLIED   MECHANICS. 


^.rr  bM  Cy° 

,*.     Difference  —   — —  I    yzdy : 
/  Jyi 


and  this  is  the  total  horizontal  force  tending  to  slide  the  piece 
AabB'on  the  face  ab. 

Area  of  face  ab,  if  #,  is  its  width,  is 


therefore  intensity  of  shear  at  a  is  approximately 

&M  C?° 

-r  I  yzJy 


or  exactly  (by  passing  to  the  limit) 

/dM\ 

dM 
And,  observing  that  F  =  -T-,  this  intensity  reduces  to 


(0 

We  may  reduce  this  expression  to  another  form  by  observ- 
ing, that,  if  yz  represent  the  distance  from  c  to  the  centre  of 
gravity  of  area  Aa,  and  A  represent  its  area,  we  have 

/Vo 

J    yzdy=y2A; 

therefore  intensity  of  shear  (at  distance  j/j  from  neutral  axis)  at 
point  a  = 

£(-M}-  (a) 

This  may  be  expressed  as  follows :  — 


LONGITUDINAL   SHEARING   OF  BEAMS.  321 

Divide  the  shearing-force  at  the  section  of  the  beam  under 
consideration,  by  the  product  of  the  moment  of  inertia  of  the 
section  and  its  width  at  the  point  where  the  intensity  of  the 
shearing-force  is  desired,  and  multiply  the  quotient  by  the  statical 
moment  of  the  portion  of  the  cross-section  between  the  point  in 
question  and  the  outer  fibre  ;  this  moment  being  taken  about  the 
neutral  axis.  The  result  is  the  required  intensity  of  shear. 

The  last  factor  is  evidently  greatest  at  the  neutral  axis ; 
hence  the  intensity  of  the  shearing-force  is  greatest  at  the 
neutral  axis. 

LONGITUDINAL  SHEARING  OF   RECTANGULAR  BEAMS. 

For  rectangular  beams,  we  have 

th* 

/=-,  *,  =  *. 

Hence  formula  (2)  becomes 


^)-  (3) 

For  the  intensity  at  the  neutral  axis,  we  shall  have,  therefore, 

I2F /h  bh\        3  F 
b*h?>  \4   2  /        2  bh? 

since  for  the  neutral  axis  we  have 

h  bh 

va  =  -     and     A  =  — . 
4  2 

EXAMPLES. 

i .  What  is  the  intensity  of  the  tendency  to  shear  at  the  neutral  axis 
of  a  rectangular  4-inch  by  1 2-inch  beam,  of  14  feet  span,  loaded  at  the 
middle  with  5000  (bs. 


322 


APPLIED   MECHANICS. 


2.  What  is  that  of  the  same  beam  at  the  neutral  axis  of  the  cross- 
section  at  the  support,  when  the  beam  has  a  uniformly  distributed  load 
of  T  2000  Ibs. 

3.  What  is  that  of  a  9-inch  by  14-inch  beam,  20  feet  span,  loaded 
with  15000  Ibs.  at  the  middle. 


§  206.  Strength  of  Hooks.  —  The  following  is  the  method 
to  be  pursued  in  determining  the  stresses  in  a 
hook  due  to  a  given  load  ;  or,  vice  versa,  the 
proper  dimensions  to  use  for  a  given  load. 

Suppose  (Fig.  233)  a  load  hung  at  E;  the 
load  being  P,  and  the  distances 
AB  —  n\ 


OF=yy 

O  being  the  centre  of  gravity  of  this 
section,  conceive  two  equal  and  opposite 
forces,  each  equal  and  parallel  to  P,  acting 
at  O. 

Let  A  =  area  of  section,  and  let  7  =  its 
moment  of  inertia  about  CD  (BCDF  represents  the  section 
revolved  into  the  plane  of  the  paper)  ;  then  — 

i  °.  The  downward  force  at  O  causes  a  uniformly  distributed 
stress  over  the  section,  whose  intensity  is 


2°.  The  downward  force  at  E  and  the  upward  force  at  O 

constitute  a  couple,  whose  moment  is 


and  this  is  resisted,  just  as  the  bending-moment  in  a  beam,  by 
a  uniformly  varying  stress,  producing  tension  on  the  left,  and 
compression  on  the  right,  of  CD. 


COLUMNS.  323 


If  we  call  p^  the  greatest  intensity  of  the  tension  due  to 
this  bending-moment,  viz.,  that  at  B,  we  have 


and  if  /3  denote  the  greatest  intensity  of  the  compression  due 
to  the  bending  moment,  viz.,  that  at  F,  we  have 


therefore  the  actual  greatest  intensity  of  the  tension  is 


and  this  must  be  kept  within  the  working  strength  if  the  load 
is  to  be  a  safe  one  ;  and  so  also  the  actual  greatest  intensity 
of  the  compression,  viz.,  that  at  F,  is,  when/,  >/,, 

,-,  _,  -A*+*)y,     ^ 
A  -A     A-          7  ^, 

which  must  be  kept  within  the  working  strength  for  compression. 

§  207.  Strength  of  Columns.  —  The  formulae  most  commonly 
employed  for  the  breaking-strength  of  columns  subjected  to  a  load 
whose  resultant  acts  along  the  axis  have  been,  until  recently, 
the  Gordon  formulae  with  Rankine's  modifications,  the  so-called 
Euler  formulae,  and  the  avowedly  empirical  formulae  of  Hodg- 
kinson.  These  formulae  do  not  give  results  which  agree  with 
those  obtained  from  tests  made  upon  such  full-size  columns  as 
are  used  in  practice. 

The  deductions  of  the  first  two  are  not  logical,  certain  assump- 
tions being  made  which  are  not  borne  out  by  the  facts. 

When  a  column  is  subjected  to  a  load  which  strains  any  fibre 
beyond  the  elastic  limit,  the  stresses  are  not  proportional  to  the 
strains,  and  hence  there  can  be  no  rational  formula  for  the  break- 
ing-load. 

Hence,    all  formulae  for  the  breaking-load  are,  of  necessity 


324  APPLIED    MECHANICS. 

empirical,  and  depend  for  their  accuracy  upon  their  agreement 
with  the  results  of  experiments  upon  the  breaking-strength  of 
such  full-size  columns  as  are  used  in  practice. 

Nevertheless,  the  ordinary  so-called  deductions  of  the   Gor- 
don, and  of  the  so-called  Euler  formula?  will  be  given  first. 

§  208.  Gordon's  Formulae  for  Columns. — (a)  Column  fixed  in 
fc  Direction  at  Both  Ends. — Let  CAD  be  the  central  axis  of  the 
column,  P  the  breaking-load,  and  v  the  greatest  deflection,  AB. 
Conceive  at  A  two  equal  and  opposite  forces,  each  equal  to  P; 
then— 

i°.  The  downward  force  at  A  causes  a  uniformly  distributed 
stress  over  the  section,  of  intensity, 

^P_ 

[D        2°.  The  downward  force  at  C  and  the  upward  force  at  A 
Fio.234.  constitute  a  bending  couple  whose  moment  is 

M=Pv. 
If  p2=  the  greatest  intensity  of  the  compression  due  to  this  bending, 


where  7=  distance  from  the  neutral  axis  to  the  most  strained  fibre  of 
the  section  at  A.     Then  will  the  greatest  intensity  of  stress  at  A  be 


and,  since  P  is  the  breaking-load,  p  must  be  equal  to  the  breaking- 
strength  for  compression  per  square  inch=/'. 

(i) 

where  p=  smallest  radius  of  gyration  of  section  at  A. 

Thus  far  the  reasoning  appears  sound;    but  in  the  next  step  it  is 
assumed  that 


GORDON'S  FORMULA  FOR   COLUMNS. 


325 


where  c  is  a  constant  to  be  determined  by  experiment.     Hence,  sub- 
stituting this,  and  solving  for  P, 

P  =      fA  ,9,  (2) 


which  is  the  formula  for  a  column  fixed  in  direction  at  both  ends. 

(b)  Column  hinged  at  the  Ends. — It  is  assumed  that  the  points  of 
inflection  are  half-way  between  the  middle  and  the  ends,  and    -jr~ 
hence  that,  by  taking  the  middle  half,  we  have  the  case  of  bending 
of  a  column  hinged  at  the  ends  (Fig.  235).     Hence,  to  obtain 
the  formula  suitable  for  this  case,  substitute,  in  (2),  2/  for  /,  and 

we  obtain 

— * — 

FIG.  235. 

"          M  <3) 


(c)  Column  fixed  at  One  End  and  hinged  at  the  Other  (Fig.  236). — 
~~r    In  this  case  we  should,  in  accordance  with  these  assumptions, 
take  J  of  the  column  fixed  in  direction  at  both  ends;   hence,  to 
obtain  the  formula  for  this  case,  substitute,  in  (2),  J/  for  /,  and 

we  thus  obtain 

j- 1 

(4) 


i6/2  ' 
gcp2 


FIG.  236. 


>.  236. 

Rankine  gives,  for  values  of  /  and  c,  the  following,  based  upon 
Hodgkinson's  experiments: 


f 
(Ibs.  per  sq.  in.). 

c. 

Wrought-iron      

36000 

36000 

Cast-iron 

80000 

Dry  timber                . 

72OO 

3OOO 

326 


APPLIED    MECHANICS. 


§  2o8a.  So-called  Euler  Formulae  for  the  Strength  of 
Columns. 

(a)  Column  fixed  in  Direction  at  One  End  only,  which  bends,  as 
shown  in  the  Figure, 

i°.  Calculate  the  breaking-load  on  the  assumption  that  the  column 
will  give  way  by  direct  compression.  This  will  be 

PI=/A,  CO 

where  /"=  crushing-strength  per  square  inch,  and  A  =  area  of  cross- 
section  in  square  inches. 

2°.  Calculate  the  load  that  would  break  the  column  if  it  were  to 
give  way  by  bending,  by  means  of  the  following  formula : 

f,  = 

where  E=  modulus  of  elasticity  of  the  material,  7=  smallest  moment 
of  inertia  of  the  cross-section,  and  /=  length  of  column. 

Then  will  the  actual  breaking-strength,  according  to  Euler,  be  the 
smaller  of  these  two  results. 

To  deduce  the  latter  formula,  assume  the  origin  at  the 
upper  end,  and  take  x  vertical  and  y  horizontal. 

Let  p=  radius  of  curvature  at  point   (x,  y),  and  let 
3/=bending-moment  at  the  same  point. 

Then   we  have,   with   compression  plus    and   tension 
minus, 


PIG.  as?. 


M_ 
El 


Py 

El' 


(3) 


But 


d*y        P 

"dx*  =  "El*' 


dy    d*y  ,          P  f  dy, 
JL  .  -J-dx  —  —  I  y-~dx 
dx    dx*  EIJJdx 


dx 


^\    =  --^y2  4-  c; 


and,  since  for  y 


Er 

~dx 


EULER  FORMULA  FOR  STRENGTH  OF  COLUMNS.      327 


dy 

J 


•    y 

•'•    Sm    -a  = 
And  since,  when  x=o,  y=©>  •"•    c=o,  we  have 


When  y=a,  x=l\  hence,  substituting  in  (5),  and  solving  for  P, 

>-(=)•« 

(b)  Column  hinged  at  Both  Ends  (Fig.  235). 
i°.  For  the  crushing-load, 

P~/A. 
2°.  For  the  breaking-load  by  bending,  put  1/2  for  I  in  (6) ;  hence 

(7) 

(c)  Column  fixed  in  Direction  at  One  End,  and  hinged  at  the  other 
<Fig.  236). 

i°.  For  the  crushing-load, 

2°.  For  the  breaking-load  by  bending,  put  //3  for  /  in  (6) ;  hence 


((/)  Column  fixed  in  Direction  at  Both  Ends  (Fig.  234). 
i°.  For  the  crushing-load, 

P.-/4. 
2°.  For  the  breaking-load  by  bending, 


:his  being  obtained  from  (2)  or  (6)  by  substituting  1/4  for  /. 


328  APPLIED   MECHANICS. 

(e)  In  order  to  ascertain  the  length  wnere  incipient  flexure  occurs, 
according  to  this  theory  we  should  place  the  two  results  equal  to  each 
other,  and  from  the  resulting  equation  determine  /.  We  should  thus 
obtain,  for  the  three  cases  respectively, 

<«)  /-  4/,  (10) 


(r)  l= 

Hence  all  columns  whose  length  is  less  than  that  given  in  these 
formulae  will,  according  to  Euler,  give  way  by  direct  crushing;  and 
those  of  greater  length,  by  bending  only. 

§  209.  Hodgkinson's  Rules  for  the  Strength  of  Columns. 
—  Eaton  Hodgkinson  made  a  large  number  of  tests  of  small  columns, 
especially  of  cast-iron,  and  deduced  from  these  tests  certain  empirical 
formulas.  The  strength  of  pillars  of  the  ordinary  sizes  used  in  practice 
has  been  computed  by  means  of  Hodgkinson's  formulae,  and  tabulated 
by  Mr.  James  B.  Francis;  and  we  find  in  his  book  the  following  rules 
for  the  strength  of  solid  cylindrical  pillars  of  cast-iron,  with  the  ends 
flat,  i.e.,  "finished  in  planes  perpendicular  to  the  axis,  the  weight 
being  uniformly  distributed  on  these  planes": 

For  pillars  whose  length  exceeds  thirty  times  their  diameter, 

^=99318^,  (.) 

where  D=  diameter  in  inches,  /=  length  in  feet,  W=  breaking-  weight 
in  Ibs. 

If,  on  the  other  hand,  the  length  does  not  exceed  thirty  times  the 
diameter,  he  gives,  for  the  breaking-weight,  the  following  formula: 


where  W=  breaking-  weight  that  would  be  derived  from  the  preceding 
formula,  W'=  actual  breaking-  weight, 


BREAKING-LOAD    OF  FULL-SIZE   COLUMNS  329 

For  hollow  cast-iron  pillars,  if  D= external  diameter  in  inches,  d= 
internal  diameter  in  inches,  we  should  have,  in  place  of  (i), 

7*.5S_^  (^ 


/'•? 

and  in  place  of  (3), 

c  =  i 0080 i 

4 

For  very  long  wrought-iron  pillars,  Hodgkinson  found  the  strength 
to  be  1.745  times  that  of  a  cast-iron  pillar  of  the  same  dimensions;  but, 
for  very  short  pillars,  he  found  the  strength  of  the  wrought-iron  pillar 
very  much  less  than  that  of  the  cast-iron  one  of  the  same  dimensions. 
With  a  length  of  30  diameters  and  flat  ends,  the  wrought-ir  on  exceeded 
the  cast-iron  by  about  10  per  cent. 

§  210.  Breaking-load  of  Full-size  Columns. — The  tests 
made  upon  full-size  columns  are  not  as  many  as  would  be  desir- 
able. The  details  will  be  given  in  Chapter  VII,  but  a  few  of  the 
empirical  formulae  which  represent  their  results  will  be  given  here. 

If  P  =  breaking-load,  A=  area  of  smallest  section,  /  =  length 
of  column,  p  =  least  radius  of  gyration  of  section,  and  }e  =  crushing- 
strength  of  the  material  per  unit  of  area,  it  will  be  found  that  for 
values  of  I/ p  less  than  a  certain  amount,  the  column  remains 
straight,  and  the  breaking-load  may  be  computed  by  means  01 
the  formula  P  =  fcA . 

For  greater  values  of  l/p,  the  breaking-load  is  smaller  than  that 
given  by  this  formula,  and  may  be  computed  by  mean;.;  of  the 
formula  P  =  fA, 

by  using  for  /  a  value  smaller  than  fc,  this  value  varying  with  the 
value  of  l/pt  and  being  determined  empirically  from  the  results  of 
tests  of  full-size  columns. 

(a)  In  the  case  of  cast-iron  columns  no  tests  have  been  made 
of  full-size  columns  of  the  second  class,  while  those  made  upon 
the  first  class  indicate  that  the  value  of  }c  suitable  for  use  in 
practice  is  from  25,000  to  30,000  Ibs.  per  square  inch. 

(b)  In  the  case  of  wrought-iron  columns,  the  tests  of  the  first 
class  indicate  that  the  value  of  }c  suitable  for  use  in  practice  is 
from  30,000  to  35,000  Ibs.  per  square  inch. 


330  APPLIED  MECHANICS 

(c)  In  the  case  of  wrought-iron  columns  of  the  second  class, 
the  formula  of  Mr.  C.  L.  Strobel  for  bridge  columns  with  either 
flat  or  pin  ends,  when  l/p>  90,  is 

p       A  * 

—  =  46000-125-. 

A  p 

On  the  other  hand,  those  recommended  by  Prof.  J.  Sonde- 
ricker,  of  which  the  first  was  devised  by  Mr.  Theodore  Cooper, 
are  as  follows: 

(a)  For  Phoenix  columns  with  flat  ends  l/p  >  So, 
P_         36000 
A 


18000 

For  lattice  columns  with  pin-ends  and  l/p>6o, 
P_=        340QO 
A 


12000 

(7-)  For  solid  web,  square,  or  box  columns  with  flat  ends,  and 
l/P>8o, 

P  33000 

Z=~(//fl-8o)2' 

10000 

($)  For  solid  web,  square,  or  box  columns  with  pin-ends,  and 
l/P>6o, 

P  _       31000 

A  (l/p  -60)* 


6000 

The  number  of  tests  that  have  been  made  upon  full-size  steel 
columns  is  very  small,  hence  no  formulae  will  be  given  here,  but 
the  subject  will  be  discussed  in  Chapter  VII.  The  number  of 
tests  that  have  been  made  upon  full-size  timber  columns  is  con- 
siderable, but  this  subject  will  also  be  discussed  in  Chapter  VII. 

§  211.  Columns  subjected  to  Loads  which  do  not  Strain 
any  Fibre  beyond  the  Elastic  Limit.  —  Under  this  head  will 
be  discussed,  first,  the  mode  of  determining  the  greatest  fibre 


THEORY  OF   COLUMNS.  331 

stress  in  a  straight  column  subjected  to  an  eccentric  load,  and, 
secondly,  the  general  theory  of  columns. 

(a)  Straight  column,  under  eccentric  load.  —  Let  O'  be  the 
centre  of  gravity  of  the  lower  section,  and  let  A'O'  =  x0,  where 
A'  is  the  point  of  application  of  the  resultant  of  the 
eccentric  load.  Conceive  two  equal  and  opposite 
forces  at  O',  each  equal  and  parallel  to  P.  Then  we 
have: 

i°.  Downward  force  along   OOr  causes   uniform 

P 

stress  of  intensity  p\  =  -r  . 

2°.  The  other  two  form  a  couple  whose  moment 
is  Px0,  and  the  greatest  intensity  of  the  stress  due  to 

this   couple   is   p2=  -  r  —  >  where   a  =  O'B'.    Hence, 
FIG.  238.  1 

the  greatest  intensity  of  the  stress  is 
P     Px0a 


and  this  should  be  kept  within  the  limits  of  the  working-strength. 

(b)  Theory  of  columns.  —  The  theory  of  columns  is  that  of 
the  Inflectional  Elastica,  and  is  explained  in  several  treatises, 
among  which  is  that  of  A.  E.  H.  Love  on  the  Theory  of  Elasticity. 
It  is  as  follows: 

Let  the  curve  OP  be  an  elastic  line,  on  which  O  is  a  point  of 
inflection.     It  follows  that  there  is  no  bending-moment  at  this 
point,    and    hence    we    may    assume 
that  at  O  a  single  force  R  acts.    Take  Y| 

the  origin  at  O,  and  axis  of  X  along  the 
line  of  action  of  the  force  R.  Let 
E\  =  modulus  of  elasticity  of  the 
material,  7  =  moment  of  inertia  of 
section  about  an  axis  through  its  centre  of  gravity,  and  perpen- 


33  2  APPLIED    MECHANICS. 

dicular  to  the  plane  of  the  curve,  (£>  =  angle  between  OX  and  the 
tangent  at  any  point  P  whose  coordinates  are  x  and  y,  a  =  value 
of  <j)  at  point  O,  r=  radius  of  curvature  of  the  curve  at  P,  s  — 
length  of  arc  OP,  /  =  length  of  one  bay,  i.e.,  measured  from  O  to 

the  next  point  of  inflection,  0=T>  -4=  area  of   section,  p  =  -*rr> 


R 

=     ' 


Then  we  have  for  any  such  elastic  line,  when  compressions  are 
plus  and  tensions  minus, 

i     M_ 
p    ET 

Moreover,  since  —  =  —  3—  and  M  =  Ry,  we  have,  for  a  column 
p         ds 

d6          R 
of  the  same  cross-section  throughout   its  length,   ~y~=~£~7>'> 

ID 

where  the  quantity  j^j  is  a  constant. 
By  differentiation  we  obtain 

R  dy         R 


Integrating,  and  observing  that  at  O,  ~^J=O»  an^  <i>=a> 


obtain 


The  integration  of  this  equation  requires  the  use  of  elliptic 
integrals,  hence  only  the  results  will  be  given  here. 


THEORY   OF  COLUMNS. 


332* 


They  are  : 


(2) 


and 


(3) 


(4) 


where  E  denotes  the  elliptic  integral  of  the  second  kind,  and  K 
the  complete  elliptic  integral  of  the  first  kind. 

Moreover,  for  the  determination  of  the  load  R,  we  obtain 
from  equation  (4) 


K=- 


and  hence 


4K2 


(6) 


From  these  equations,  we  can,  by  using  a  table  of  elliptic 
functions,  deduce  the  following  results  for  the  coordinates  of  points 
on  the  inflectional  elastica,  for  various  values  of  a : 


a 

5 

T 

X 

/ 

y 
I 

10° 

o.oo 

o  .  oooo 

0  .  OOOO 

0.25 

0.50 

0.2476 

o  .  4962 

o  .0392 
°-°554 

20° 

o.oo 

o  .  oooo 

0  .  OOOO 

0.25 
o.  50 

0.2376 

o  .  4849 

0.0773 

o.  1079 

30° 

o.oo 

0  .  OOOO 

0  .  OOOO 

0.25 

0.50 

0.2224 
o  .  4662 

0.1135 

o.  1620 

Moreover,  these  results  agree  with  those  which  we  obtain  by 


APPLIED  MECHANICS. 


experiment,  and  thus  we  can,  by  making  use  of  our  calculations, 
compute  the  load  required  to  produce  a  given  elastica,  determined 
by  the  slope  at  the  points  of  inflection,  which,  in  the  case  of  pin- 
ended  columns,  are  at  the  ends,  and,  in  the  case  of  columns  fixed 
in  direction  at  the  ends,  are  half-way  between  -the  middle  and  the 
ends. 

All  this  can  be  done,  and  can  be  verified  by  experiment, 
provided  that  the  load  is  not  so  great  that  any  fibre  is  strained 
beyond  the  elastic  limit  of  the  material,  and  provided  the  value  of 
l/p  is  not  so  small  that  the  curvilinear  form  is  unstable. 

For  smaller  values  of  l/p  the  only  stable  form  is  a  straight  line, 
and  the  column  does  not  bend. 

To  ascertain  the  least  value  of  l/p  for  which  a  curved  form  is 
stable,  observe  that  K  cannot  be  less  than  71/2,  and  since  this  cor- 
responds to  one  bay,  and  hence  to  the  case  of  a  pin-ended  column, 
we  have  in  that  case,  by  substituting  n/2  for  K  in  equation  (6)  , 

7T2 


n 

and,  since  I=AfP    and     -7=<7, 


we  have  for  the  line  of  demarcation  between  the  straight  and 
curved  form  in  a  pin-ended  column 


-;  (7) 

and  for  that  in  the  case  of  a  column  fixed  in  direction  at  the  ends 


As  an  example,  if  a=  10,000  and  £1=30,000,000  we  should 
find  that  a  pin-ended  column  would  not  bend  unless  l/p  were 
greater  than  172,  and  that  a  column  fixed  in  direction  at  the  ends 


STRENGTH  OF  SHAFTING.  333 

would  not  bend  unless  l/p  were  greater  than  344.  Columns  with 
smaller  values  of  l/p  would  remain  straight  when  the  resultant  of 
the  load  acts  along  the  axis,  and  no  fibre  is  strained  beyond  the 
elastic  limit. 

§  212.  Strength  of  Shafting.  —  The  usual  criterion  for  the 
strength  of  shafting  is,  that  it  shall  be  sufficiently  strong  to 
resist  the  twisting  to  which  it  is  exposed  in  the  transmission  of 
power. 

Proceeding- in  this  way,  let  EF  (Fig.  239)  be  a  shaft,  AB  the 
driving,  and  CD  the  following,  pulley. 
Then,  if  two  cross-sections  be  taken 
between  these  two  pulleys,  the  por- 
tion of  the  shaft  between  these  two 
cross-sections  will,  during  the  trans- 
mission of  power,  be  in  a  twisted  con-  F 

riG.  239. 

dition ;  and  if,  when  the  shaft  is  at 

rest,  a  pair  of  vertical  parallel  diameters  be  drawn  in  these  sec- 
tions, they  will,  after  it  is  set  in  motion,  no  longer  be  parallel, 
but  will  be  inclined  to  each  other  at  an  angle  depending  upon 
the  power  applied.  Let  GH  be  a  section  at  a  distance  x  from 
O,  and  let  KI  be  another  section  at  a  distance  x  -f-  dx  from  O. 
Then,  if  di  represent  the  angle  at  which  the  originally  parallel 
diameters  of  these  sections  diverge  from  each  other,  and  if  r  = 
the  radius  of  the  shaft,  we  shall  have,  for  the  length  of  an  arc 
passed  over  by  a  point  on  the  outside, 

rdi; 

and  for  the  length  of  an  arc  that  would  be  passed  over  if  the 
sections  were  a  unit's  distance  apart,  instead  of  dx  apart, 

rdi  _      di 
dx         dx 

This  is  called  the  strain  of  the  outer  fibres  of  the  shaft,  as  it 
is  the  distortion  per  unit  of  length  of  the  shaft. 


334  APPLIED  MECHANICS. 

In  all  cases  where  the  shaft  is  homogeneous  and  symmet- 
rical, if  i  is  the  angle  of  divergence  of  two  originally  parallel 
diameters  whose  distance  apart  is  x,  we  shall  have  the  strain, 

di         i 
v  =  r —  =  r-. 

dx        x 

This  also  is  the  tangent  of  the  angle  of  the  helix. 

A  fibre  whose  distance  from  the  axis  of  the  shaft  is  unity, 
will  have,  for  its  strain, 

dt_  =   / 

dx        x 

A  fibre  whose  distance  from  the  axis  of  the  shaft  is  p,  will  have, 
for  its  strain, 

di         i 

v  =  p— -  =  p-. 

dx        x 

Fixing,  now,  our  attention  upon  one  cross-section,  GH,  we  have 
that  the  strain  of  a  fibre  at  a  distance  p  from  the  axis  (p  varying, 
and  being  the  radius  of  any  point  whatever)  is 


where  -  is  a  constant  for  all  points  of  this  cross-section. 

X 

Hence,  assuming  Hooke's  law,  "  Ut  tensio  sic  vis"  we  shall 
have,  if  C  represent  the  shearing  modulus  of  elasticity,  that  the 
stress  of  a  fibre  whose  distance  from  the  axis  is  p,  is 


which  quantity  is  proportional  to  p,  or  varies  uniformly  from  the 
centre  of  the  shaft. 

The  intensity  at  a  unit's  distance  from  the  axis  is 


•0- 


STRENGTH  OF  SHAFTING.  335 

and  if  we  represent  this  by  a,  we  shall  have  for  that  at  a  dis- 
tance p  from  the  axis, 


Hence  we  shall  have  (Fig,  240),  that,  on  a  small 
area,  V  J 

dA  =  dp(PdB)  _  pdpdO,  ^^ 

the  stress  will  be 

pdA  =  apdA  =  ap2dpd9. 

The  moment  of  this  stress  about  the  axis  of  the  shaft  is 

ppdA  =  apzdA  =  ap^dpdO, 

and  the  entire  moment  of  the  stress  at  a  cross-section  is 
afp*dA  =  affpidpdO  =  al, 

where  /  =  fp2dA  is  the  moment  of  inertia  of  the  section  about 
the  axis  of  the  shaft. 

This  moment  of  the  stress  is  evidently  caused  by,  and  hence 
must  be  balanced  by,  the  twisting-moment  due  to  the  pull  of  the 
belt.  Hence,  if  M  represent  the  greatest  allowable  twisting- 
moment,  and  a  the  greatest  allowable  intensity  of  the  stress  at 
a  unit's  distance  from  the  axis,  we  shall  have 

M  =  al  =  -  /. 

P 

If  /  is  the  safe  working  shearing-strength  of  the  material 
per  square  inch,  we  shall  have  /  as  the  greatest  safe  stress  per 
square  inch  at  the  outside  fibre,  and  hence 

M=- I 
r 

will  be  the  greatest  allowable  twisting-moment. 


33^  APPLIED   MECHANICS. 

For  a  circle,  radius  rt 

2          "         ~  *  ~~2~  ~J  ~i6~* 
For  a  hollow  circle,  outside  radius  rv  inside  radius  rM 


Moreover,  if  the  dimensions  of  a  shaft  are  given,  and  the 
actual  twisting-moment  to  which  it  is  subjected,  the  stress  at  a 
fibre  at  a  distance  p  from  the  axis  will  be  found  by  means  of  the 
formula 


The  more  usual  data  are  the  horse-power  transmitted  and 
the  speed,  rather  than  the  twisting-moment. 

If  we  let  P  =  force  applied  in  pounds  and  R  =  its  leverage 
in  inches,  as,  for  instance,  when  P  =  difference  of  tensions  of 
belt,  and  R  =  radius  of  pulley,  we  have 


and    if    HP  —  number    of    horses-power    transmitted,    and 
N  =  number  of  turns  per  minute,  then 


TT  T)    _ \  •*•'    *•_  /_    . 

12    X    33000  ' 
12    X    -l^OOoIfP 

—    ^r^r. ^—    Jyi 

271 N 

EXAMPLE. 

Given  working-strength  for  shearing  of  wrought-iron  as  10000 
Ibs.  per  square  inch  ;  find  proper  diameter  of  shaft  to  transmit 
2o-horse  power,  making  100  turns  per  minute. 


TRANSVERSE  DEFLECTION  OF  SHAFT.  337 


Mp 
Angle  of  Torsion. — From  the  formula,  page  336,  p~  —=- % 


combined  with 


we  have 


=  ap  =  Cp-, 

oc 


.  _  MX 
"  ~' 


which  gives  the  circular  measure  of  the  angle  of  divergence  of 
two  originally  parallel  diameters  whose  distance  apart  is  x  ;  the 
twisting-moment  being  M,  and  the  modulus  of  shearing  elas- 
ticity of  the  material,  C. 

EXAMPLES. 

1.  Find  the  angle  of  twist  of  the  shaft  given  in  example  i,  §  212, 
when  the  length  is  10  feet,  and  C  =  8500000. 

2.  What  must  be  the  diameter  of  a  shaft  to  carry  80  horses-power, 
with  a  speed  of  300  revolutions  per  minute,  and  factor  of  safety  6,  break- 
ing shearing-strength  of  the  iron  per  square  inch  being  50000  Ibs. 

§  213.  Transverse  Deflection  of  Shafts.  —  In  determining 
the  proper  diameter  of  shaft  to  be  used  in  any  given  case,  we 
ought  not  merely  to  consider  the  -resistance  to  twisting,  but 
also  the  deflection  under  the  transverse  load  of  the  belt-pulls, 
weights  of  pulleys,  etc.  This  deflection  should  not  be  allowed 
to  exceed  y^-  of  an  inch  per  foot  of  length.  Hence  the  de- 
flection should  be  determined  in  each  case. 

The  formulae  for  computing  this  deflection  will  not  be  given 
here,  as  the  methods  to  be  pursued  are  just  the  same  as  in  the 
case  of  a  beam,  and  can  be  obtained  from  the  discussions  on 
that  subject. 


APPLIED   MECHANICS. 


§  214.  Combined  Twisting  and  Bending.  —  The  most  com- 
mon case  of  a  shaft  is  for  it  to  be  subjected  to  combined  twisting 
and  bending.  The  discussion  of  this  case  involves  the  theory 
of  elasticity,  and  will  not  be  treated  here  ;  but  the  formulae  com- 
monly given  will  be  stated,  without  attempt  to  prove  them  until 
a  later  period.  These  formulae  are  as  follows  :  — 
Let  Ml  =  greatest  bending-moment, 

M2  =  greatest  twisting-moment, 

r     =  external  radius  of  shaft, 

/     =  moment  of  inertia  of  section  about  a  diameter, 

TTf4 

for  a  solid  shaft  /  =  —  —  , 
4 

f     =  working-strength  of  the  material  =  greatest  al- 

lowable stress  at  outside  fibre  ; 
then 

i°.  According  to  Grashof, 


/=  LjfJ/i  +  fVJ/x2  +  M*\.  (i) 

2°.  According  to  Rankine, 


/ =£  j  M,  +  \!M*  +  M;  j .  (2) 

§  215.  Springs. — The  object  of  this  discussion  is  to  enable  us 
to  answer  the  following  three  questions :  (a)  Given  a  spring, 
to  determine  the  load  that.it  can  bear  without  producing  in  the 
metal  a  maximum  fibre  stress  greater  than  a  given  amount. 
(&)  Given  a  spring,  to  determine  its  displacement  (elongation, 
compression,  or  deflection)  under  any  given  load,  (c)  Given  a 
load  P  and  a  displacement  &t  ;  a  spring  is  to  be  made  of  a 
given  material  such  that  the  load  P  shall  produce  the  displace- 
ment 6I ,  and  that  the  metal  shall  not,  in  that  case,  be  subjected 
to  more  than  a  given  maximum  fibre  stress.  Determine  the 
proper  dimensions  of  the  spring. 


SPRINGS.  339 


There  are  practically  only  two  cases  to  be  considered  as  far 
as  the  manner  of  resisting  the  load  is  concerned.  In  the  first, 
the  spring  is  subjected  to  transverse  stress,  and  is  to  be  calcu- 
lated by  the  ordinary  rules  for  beams.  In  the  second,  the 
spring  is  subjected  to  torsion,  and  the  ordinary  rules  for  re- 
sistance to  torsion  apply.  It  is  true  that  in  most  cases  where 
the  spring  is  subjected  to  torsion  there  is  also  a  small  amount 
of  transverse  stress  in  addition  to  the  torsion  ;  but  in  a  well- 
made  spring  this  transverse  stress  is  of  very  small  amount,  and 
we  may  neglect  it  without  much  error. 

We  will  begin  with  those  cases  where  the  spring  is  subjected 
to  torsion,  and  for  all  cases  we  shall  adopt  the  following  nota 
tion  : 

P  =  load  on  spring  producing  maximum  fibre  stress/; 

f  =  greatest  allowable  maximum  fibre  stress  for  shearing ; 

C  =  shearing  modulus  of  elasticity ; 

x  =  length  of  wire  forming  the  spring ; 
Ml  =  greatest  twisting  moment  under  load  P\ 

L  =  any  load  less  than  the  limit  of  elasticity ; 
M  =  twisting  moment  under  this  load  ; 

p  =  maximum  fibre  stress  under  load  L ; 

p  —  distance  from  axis  of  wire  to  most  strained  fibre  ; 

/  =  moment  of  inertia  of  section  about  axis  of  wire ; 

z'j  =  angle  of  twist  of  wire  under  load  P- 
i  =  angle  of  twist  of  wire  under  load  L ; 

V  =  volume  of  spring  ; 

#j  =  displacement  of  point  where  load  is  applied  when  load 
isP; 

d  —  displacement  of  point  where  load  is  applied  when  load 
isZ. 

Then  from  pages  335  and  337  we  obtain  the  following  four 
formulae : 

*•=£/,  (i) 


34O  APPLIED   MECHANICS. 


MX 
'=C7' 


(3) 


These  four  formulae  will  enable  us  to  solve  all  the  cases  of 
springs  subjected  to  torsion  only.  Moreover,  in  the  cases 
which  we  shall  discuss  under  this  head,  the  wire  will  have  either 
a  circular  or  a  rectangular  section  :  in  the  former  case  we  will 
denote  its  diameter  by  d,  and  we  shall  then  have 

net*  d 

/=  --          and         p  =  —  ; 
32  2 

while  in  the  latter  case  we  will  denote  the  two  dimensions  of 
the  rectangle  by  b  and  h,  respectively,  and  we  shall  then  have 


We  will  now  proceed  to  determine  the  values  of  P,  #,  Sl  ,  and 
V  in  each  of  the  following  four  cases,  all  of  which  are  cases  of 
torsion  : 

CASE  i.  Simple  round  torsion  wire.  —  Let  AB,  the  leverage 
of  the  load  about  the  axis,  be  R  ;  then  we  shall  have 

M  =  LR,        M,  =  PR  ; 
and  we  readily  obtain  from  the  formulae  (i),  (2),  (3),  and  (4) 


^\ 


,f    C-  <" 


(7) 


SPRINGS. 


341 


and  from  these  we  readily  obtain 


(8) 


CASE  2.'  Simple  rectangular  torsion  wire. — In  this  case  we 
readily  obtain 


(9) 


«,        D.  ,     . 

=  ™=ri'    (IC 


>=*'>  =  "' 


CASES  3  and  4.  Helical  springs  made  of  round  and  of  rec- 
tangular wire  respectively. — A  helical  spring  may  be  used  either 
in  tension  or  in  compression.  In  either  case  it  is  important 
that  the  ends  should  be  so  guided  that  the  pair  of  equal  and 
opposite  forces  acting  at  the  ends  of  the  spring  should  act  ex- 
actly along  the  axis  of  the  spring. 

This  is  of  especial  importance  when  the  spring  is  used  for 
making  accurate  measurements  of  forces,  as  in  the  steam-en- 
gine indicator,  in  spring  balances,  etc. 

Moreover,  it  is  generally  safer,  as  far  as  accuracy  is  con- 
cerned, to  use  a  helical  spring  in  tension  rather  than  in  com- 
pression, as  it  is  easier  to  make  sure  that  the  forces  act  along 


342  APPLIED  MECHANICS. 

the  axis  in  the  case  of  tension  than  in  the  case  of  compres- 
sion. 

Whichever  way  the  spring  is  used,  however,  provided  only 
the  two  opposing  forces  act  along  the  axis  of  the  spring,  the 
resistance  to  which  the  spring  is  subjected  is  mainly  torsion, 
inasmuch  as  the  amount  of  bending  is  very  slight. 

This  bending,  however,  we  will  neglect,  and  will  compute 
the  spring  as  a  case  of  pure  torsion,  the  same  notation  being 
used  as  before,  except  that  we  will  now  denote  by  R  the  radius 


of  the  spring,  and  we  shall  have 

M  =  LR,    M,  = 

and  now  formulae  (5),  (6),  (7),  and  (8)  become  applicable  to  a 
spring  made  of  round  wire,  and  formulae  (9)  and  (10),  (n)  and 
(12),  to  one  made  of  rectangular  wire. 

We  must  bear  in  mind,  however,  that  x  denotes  the  length 
of  the  wire  composing  the  spring,  and  not  the  length  of  the 
spring,  d  and  dl  now  denote  the  elongations  or  compressions 
of  the  spring. 

GENERAL  REMARKS. 

By  comparing  equations  (8)  and  (12),  it  will  be  seen 
that  if  a  spring  is  required  for  a  given  service,  its  volume 
and  hence  its  weight  must  be  50  per  cent  greater  if  made 
of  rectangular  than  if  made  of  round  wire.  Again,  it  is 
evident  that  when  the  kind  of  spring  required  is  given. 


SPRINGS.  343 


and  the  values  of  C  and  f  for  the  material  of  which  it  is  to 
be  made  are  known,  the  volume  and  hence  the  weight  of 
the  spring  depends  only  on  the  product  Pdlt  and  that  as  soon 
as  P  and  d\  are  given,  the  weight  of  the  spring  is  fixed  inde- 
pendently of  its  special  dimensions.  If,  however,  we  fix  any 
one  dimension  arbitrarily,  the  others  must  be  so  fixed  as  to 
satisfy  the  equations  already  given.  Next,  as  to  the  values  to 
be  used  for  /and  C,  these  will  depend  upon  the  nature  of  the 
special  material  of  which  the  spring  is  made,  and  these  can 
only  be  determined  by  experiment.  Confining  ourselves  now 
to  the  case  of  steel  springs,  it  is  plain  that  /and  £7  should  be 
values  corresponding  to  tempered  steel. 

As  an  example,  suppose  we  require  the  weight  of  a  helical 
spring,  which  is  to  bear  a  safe  load  of  10000  Ibs.  with  a  deflec- 
tion of  one  inch,  assuming  C=  12600000  and/=  80000  Ibs. 
per  sq.  in.,  and  as  the  weight  of  the  steel  0.28  Ib.  per  cubic 
inch. 

From  formula  (8)  we  obtain 

_,        2  X  12600000  X  10000  X  i 

=39.4cu.m. 


Hence  the  weight  of  the  spring  must  be  (39.4)  (0.28)  =  II  Ibs. 
We  may  use  either  a  single  spring  weighing  1  1  Ibs.,  or  else 
two  or  more  springs  either  side  by  side  or  in  a  nest,  whose  com- 
bined weight  is  1  1  Ibs.  Of  course  in  the  latter  case  they  must 
all  deflect  the  same  amount  under  the  portion  of  the  load 
which  each  one  is  expected  to  bear,  and  this  fact  must  be 
taken  into  account  in  proportioning  the  separate  springs  that 
compose  the  nest. 

FLAT   SPRINGS. 

Let  P,  L,  V,  d,  and  d^  have  the  same  meanings  as  before, 
and  let 


344 


APPLIED  MECHANICS, 


f=  greatest  allowable  fibre  stress  for  tension  or  compres- 

sion  : 

R  =  modulus  of  elasticity  for  tension  or  compression  ; 
/=  length  of  spring; 

Ml  =  maximum  bending-moment  under  load  P ; 
M=  maximum  bending-moment  under  load  L. 

Moreover,  the  sections  to  be  considered  are  all  rectangular, 
and  we  will  let  b  =  breadth  and  h  =  depth  at  the  section 
where  the  greatest  bending-moment  acts,  the  depth  being 
measured  parallel  to  the  load. 

Then  if  /  denote  the  moment  of  inertia  of  the  section  of 
greatest  bending-moment  about  its  neutral  axis,  we  shall  have 

f=M 

12 

We  will  now  consider  six  cases  of  flat  springs,  and  will  de- 
termine P,  tf,  tfz ,  and  V  for  each  case,  and  for  this  purpose  we 
only  need  to  apply  the  ordinary  rules  for  the  strength  and  de- 
flection of  beams. 

CASE  i.  Simple  rectangular  spring,  fixed  at  one  end  and 
loaded  at  the  other. 


I3  L 
i*>£ 

I*     f 


(24) 


E 
E 


(26) 


SPRINGS. 


345 


CASE  2.  Spring-  of  uniform  depth  and  uniform  strength,  tri- 
in  plan ,  fixed  at  one  end  and  loaded  at  the  other. 


(27) 
(28) 

(29) 


(30) 


CASE  3.  Spring  of  uniform  breadth  and  uniform  strength, 
parabolic  in  elevation,  fixed  at  one  end  and  loaded  at  the 
other. 


(31) 


(33) 


(34) 


CASE  4.  Compound  wagon  spring,  made  up  of  n  simple  rec- 
tangular springs  laid  one  above  the  other,  fixed  at  one  end  and 
loaded  at  the  other. 


346 


APPLIED  MECHANICS. 


Let  the  breadth  be  b,  and  the  depth  of  each  separate  layer 

be  h.     Then 

'- 

n     bh* 
6/   /' 

(35) 

i 

N^ 

6  = 

4  /*  L 

nbh*  E* 

(30 

i 

=3=^ 

/'/ 

(37) 

i 

\ 

i 

y^  ^fi1  * 

(38) 


CASE  5.  Compound  spring  composed  of  n  triangular  springs 
laid  one  above  the  other,  fixed  at  one  end  and  loaded  at  the 
other. 


*-\r=r  ^ 

6  =  -nWLE>  <4°> 

*»=ji;  (4I) 


CASE  6.  This  case  differs  from  the  last  in  that  in  order  to 
economize  material  we  superpose  springs  of  different  lengths, 


SPRINGS. 


347 


and  make  them  of  such  a  shape  that  by  the  action  of  a  single 
force  at  the  free  end  they  are  bent  in  arcs  of  circles  of  nearly 
or  exactly  the  same  radius. 
The  force  P  bends  the 
lowest  triangular  piece  AA 


in 


the  arc  of  a  circle. 


length  of  this  piece  is  -. 


In  order  that  the  re- 
maining parallelopipedical 
portion  may  bend  into  an 
arc  of  the  same  circle  it  is 
necessary  that  it  should  have 

acting  on  it  a  uniform  bending-moment  throughout,  and  this 
is  attained  if  it  exerts  a  pressure  at  Al  upon  the  succeeding 
spring  equal  to  the  force  P,  and  following  this  out  we  should 
find  that  the  entire  spring  would  bend  in  an  arc  of  a  circle. 

The  values  of  P,  6,  dz ,  and  Fare  correctly  expressed  for 
this  case  by  (39),  (40),  (41),  and  (42). 

For  any  flat  springs  which  are  supported  at  the  ends  and 
loaded  at  the  middle,  or  where  two  springs  are  fastened  to- 
gether, it  is  easy  to  compute,  by  means  of  the  formulae  already 
developed,  by  making  the  necessary  alterations,  the  quantities 
P.  3  dz ,  and  V,  and  this  will  be  left  to  the  student. 

COILED    SPRINGS    SUBJECTED    TO    TRANSVERSE    STRESS. 

Three  cases  of  coiled  springs  will  now  be  given  as  shown 
in  the  figures,  and  the  values  of  P,  3,  dlt  and  Fwill  be  deter- 
mined for  each. 

In  each  of  these  cases  let  R  be  the  leverage,  of  the  load, 
and  let  GO  =  angle  turned  through  under  the  load.  Then  we 
may  observe  that  all  the  three  cases  are  cases  of  beams  sub- 
jected to  a  uniform  bending-moment  throughout  their  length, 
this  bending-moment  being  LR  for  load  L  and  PR  for  load  P. 


348 


APPLIED  MECHANICS. 


CASES  I  and  2.  Coiled  spring,  rectangular  in  section. 

f  b>i  i     \ 

^  =  i/-^>  (43) 

UP  L  ,     , 

(44) 

(45) 


f  Rl 


CASE  3.  Coiled  spring,  cir- 
cular in  section. 

f  =  -^f^>  (47) 

64  l^_L  .    . 

~*    j*    z^>  \4w 

(49) 


E 


TIME   OF   OSCILLATION    OF   A   SPRING. 


(46) 


(So) 


Since  in  any  spring  the  load  producing  any  displacement 
is  proportional  to  the  displacement,  it  follows  that  when  a 
spring  oscillates,  its  motion  is  harmonious. 


SPRINGS.  349 


Suppose  the  load  on  the  spring  to  be  Pt  and  hence  its  nor- 
mal  displacement  to  be  <S\.  Now  let  the  extreme  displacements 
on  the  two  sides  of  #,  be  #0  ,  and  the  force  producing  it  />,  so  that 
the  actual  displacement  varies  from  #x  -f-  tf0  to  <$,  —  <?0  ,  and  the 
force  acting  varies  from  P  -\-  p  to  P  —  p. 

Now.  from  the  properties  of  the  spring  we  must  have 

£=£;    /.*.  =  £*..  (so 

Moreover,  in  the  case  of  harmonic   motion  the  maximum 


value  of  the  force  acting  is  -  (see  p.   104).     But  the  load 

o 

oscillating  is  P  instead  of  W,  and  the  extreme  displacement  is 
6e  instead  of  r. 
Hence  we  have 


(52) 

«S  d 


(S3) 
Hence  the  time  of  a  double  oscillation 

(54) 

g 


35°  APPLIED   MECHANICS. 


CHAPTER  VII. 

STRENGTH    OF    MATERIALS    AS    DETERMINED    BY 
EXPERIMENT. 

§  216.  Whatever  computations  are  made  to  determine  the 
form  and  dimensions  of  pieces  that  are  to  resist  stress  and 
strain  should  be  based  upon  experiments  made  upon  the  mate- 
rials themselves. 

The  most  valuable  experiments  are  those  made  upon  pieces 
of  the  same  quality,  size,  and  form  as  those  to  which  the  results 
are  to  be  applied,  and  under  conditions  entirely  similar  to  those 
to  which  the  pieces  are  subjected  in  actual  practice. 

From  such  experiments  the  engineer  can  learn  upon  what  he 
can  rely  in  designing  any  structure  or  machine,  and  this  class  of 
tests  must  be  the  final  arbiter  in  deciding  upon  the  quality  of 
material  best  suited  for  a  given  service.  An  attempt  will  be  made 
in  this  chapter  to  give  an  account  of  the  most  important  results 
of  experiments  on  the  strength  of  materials,  and  to  explain  the 
modes  of  using  the  results. 

While  the  importance  of  making  tests  upon  full-size  pieces, 
and  of  introducing  into  the  experiments  the  conditions  of 
practice,  is  pretty  generally  recognized  to-day,  nevertheless 
there  are  some  who  have  not  yet  learned  to  recognize  the  fact 
that  attempts  to  infer  the  behavior  of  full-size  pieces  under 
practical  conditions  from  the  results  of  tests  on  small  models, 
made  under  conditions  which  are,  as  a  rule,  necessarily,  quite 
different  from  those  of  practice,  are  very  liable  to  lead  to  con- 
clusions that  are  entirely  erroneous. 


GENERAL   REMARKS.  351 

Such  a  proceeding  is  in  direct  violation  of  a  principle  that 
the  physicist  is  careful  to  observe  throughout  his  work,  viz.: 
not  to  apply  the  results  to  cases  where  the  conditions  are  essentially 
different  from  those  of  the  experiments. 

When  the  quality  of  the  material  suited  for  a  given 
service  is  known,  tests  of  the  material  furnished  must  be 
made  to  determine  its  quality.  Such  tests,  made  upon 
small  samples,  should  be  of  such  a  kind  that  there  may 
be  a  clear  understanding,  as  to  the  quality  desired,  between 
the  maker  of  the  specifications  and  the  producer.  Whenever 
possible,  standard  forms  of  specimens  and  standard  methods 
of  tests  should  be  used. 

The  determination  of  standards  is  occupying  the  at- 
tention of  the  Int.  Assoc.  for  Testing  Materials,  the  British 
Standards  Committee,  the  Am.  Soc.  for  Testing  Materials, 
and  others. 

To  ascertain  the  quality  of  the  material  tensile  tests  are  most 
frequently  employed,  their  objects  being  to  determine  the  tensile 
strength  per  square  inch,  the  limit  of  elasticity,  the  yield-point, 
the  ultimate  contraction  of  area  per  cent,  the  ultimate  elongation 
per  cent  in  a  certain  gauged  length,  and  sometimes  the  modulus 
of  elasticity. 

While  the  standard  forms  and  dimensions  will  be  given  later, 
the  following  general  classification  of  the  forms  in  use  will  be 
given  here,  viz.  • 

i°.  The  specimen  may  be  provided  with  a  shoulder  at  each 
end,  having  a  larger  sectional  area  than  the  main  body  of  the 
specimen,  the  section  of  this  being  uniform  throughout  as  shown 
in  Fig.  a,  the  latter  being  of  so  great  a  length  in  proportion  to  the 

diameter    that    the    stretch    of     i — i 

i  i  i  i  i  i  i  i  i 

the  specimen  is  not  essentially     I • 

different   from   what   it   would  FIG.  a. 

be  if  the  section  were  uniform  throughout.     The   shoulders  are, 


352  APPLIED  MECHANICS. 

of  course,  the  portions  of  the  specimen  where  the  holders  (or 
clamps)  of  the  testing-machine  are  attached. 

2°.  In  the  case  of  a  round  specimen  of  that  kind  there  may 
be  a  screw-thread  on  the  shoulders  as  shown  in  Fig.  b. 

In  the  case  of  a  brittle  material,  as 
1^  JJJII     cast-iron  or  hard  steel,  it  is  desirable  to 

use  a  holder  with  a  ball-joint,  and  to 
screw  the  specimen  into  the  holder. 

3°.  The  specimen  may  be  provided  with  a  shoulder  at  each 
end,  the  main  body  of  the  specimen  being,  however,  so  short 
in  proportion  to  the  diameter  that  the  stretch  is  essentially 
modified.  Such  a  form  is  shown  in  Fig.  c. 


FIG.  c.  FIG.  d. 

4°.  The  specimen  may  be  a  grooved  specimen  as  shown  in 
Fig.  d,  where  the  length  of  the  smallest  section  is  zero. 

5°.  The  section  of  the  specimen  may  be  uniform  through- 
out, the  length  between  the  holders  being  so  great  in  propor- 
tion to  the  diameter  that  the  stretching  of  the  fibres  is  not 
interfered  with.  This  form  of  specimen  is  shown  in  Fig.  e. 

Assume  a  specimen  of  duc- 

I  I  'tile  material,  as  mild  steel  or 

wrought-iron,  of  the  ist  or  the 
FlG-  '•  5th  shape,  subjected  to  stress 

in  the  testing-machine,  or  else  by  direct  weight,  and  suppose 
that  we  mark  off  upon  the  main  body,  i.e.,  the  parallel  section 
of  the  specimen,  a  gauged  length  of  8  or  10  inches  (preferably 
8  inches),  and  measure,  by  means  of  some  form  of  extensom- 
eter,  the  elongations  in  the  gauged  length,  corresponding  to 
the  stresses  applied  ;  then  plot  a  stress-strain  diagram  as  shown 
in  Fig.  /,  having  stresses  per  square  inch  for  abscissae,  and  the 
corresponding  strains  for  ordinates. 


GENERAL   REMARKS. 


353 


CO 

111 

5- 

2      0" 

</ 

f^-- 

£ 

'B 

"eo  -too 

Z.m 

I 

O  .002 
h 
u       o 

„  —  • 

-—•- 

**-*" 

.--*• 

^ 

-""" 

^~~- 

^-- 

~^~ 

^- 

„--• 

r^" 

•-•"" 

OC        V      2000           6000          KM 
CO 

XX)         14( 

XX)         18 
LOAD 

XX)         22 
PER  SQ. 

K)0         20000         30< 
IN, 

XX)         31000         3800C 

FIG.  /. 

We  shall  find  that  the  strains  begin  by  being  proportional 
to  the  stresses,  but  when  a  certain  stress  is  reached,  called  the 
"  limit  of  elasticity  "  or  "  elastic  limit,"  shown  at  A,  the  strains 
increase  more  rapidly  than  the  stresses,  but  the  rate  of  increase 
in  the  ratio  of  the  strain  to  the  stress  is  not  large  until  a  stress 
is  reached  called  the  "  yield-point  "  or  "stretch-limit,"  shown 
at  B,  which  is  usually  a  little  larger  than  the  elastic  limit ;  and 
then  the  rate  of  increase  of  the  ratio  of  strain  to  stress  becomes 
much  larger. 

Observe,  also,  that  if  a  small  load  be  applied  to  the  piece 
under  test,  and  then  removed,  the  deformation  or  distortion 
caused  by  the  application  of  the  load  apparently  vanishes,  and 
the  piece  resumes  its  original  form  and  dimensions  on  the 
removal  of  the  load  ;  in  other  words,  no  permanent  set  takes 
place.  When  the  load,  however,  is  increased  beyond  a  certain 
point,  the  piece  under  test  does  not  return  entirely  to  its 
original  dimensions  on  the  removal  of  the  load,  but  retains  a 
certain  permanent  set.  While  permanent  set  that  is  easily 
determined  begins  at  or  near  the  elastic  limit,  and  while  the 
permanent  sets  corresponding  to  stresses  greater  than  the 
elastic  limit  are  much  greater  than  the  corresponding  recoils, 
and  hence  form  the  greater  part  of  the  strains  corresponding 
to  such  stresses,  nevertheless  experiments  show  that  even  a 
very  small  load  will  often  produce  a  permanent  set,  and  that 
the  apparent  return  of  the  piece  to  its  original  dimensions  is, 


354  APPLIED    MECHANICS. 

in  a  number  of  cases,  only  due  to  the  want  of  delicacy  in  the 
measuring-instruments  at  our  command. 

I-  After  the  elastic  limit  and  the  yield-point  have  been  passed, 
the  ratio  of  the  strain  to  the  stress  is  much  greater  than  before, 
the  stretch  becomes  local,  with  a  local  contraction  of  area,  this 
being  due  to  the  plasticity  of  the  metal. 

Finally,  when  the  maximum  stress  is  applied,  or,  in  other 
words,  the  breaking-stress,  the  behavior  is  apparently  some- 
what different  when  the  piece  is  subjected  to  dead  weight  from 
what  it  is  when  in  a  testing-machine.  In  the  former  case, 
when  the  maximum  load  is  reached,  the  specimen  continues  to 
stretch  rapidly,  without  increase  in  the  load,  until  the  specimen 
breaks. 

In  the  case  of  the  testing-machine,  however,  the  application 
of  the  maximum  load  causes,  of  course,  the  specimen  to 
stretch,  but  this  stretch  naturally  reduces  the  load  applied,  and 
the  actual  load  under  which  the  specimen  separates  into  two 
parts  is  less,  and  often  very  considerably  less,  than  the  maxi- 
mum or  breaking  stress. 

Observe  that  the  terms  "  breaking-load  "  and  "  breaking- 
stress  "  are  always  used  to  mean  the  "  maximum  load  "  and 
"maximum  stress  "  respectively,  and  are  never  used  to  denote 
the  load  or  the  stress  under  which  the  specimen  separates  into 
two  parts  when  the  latter  differs  from  the  former. 

If  the  stretch  of  the  specimen,  as  described  above,  is  in  any 
way  interfered  with,  the  behavior  of  the  specimen  will  not  be 
a  proper  criterion  of  the  properties  of  the  material  ;  the  per- 
centage contraction  of  area  at  fracture  will  vary  with  the 
amount  of  interference  with  the  stretch,  and  hence  with  the 
proportions  of  the  specimen  ;  and  the  maximum  or  breaking 
strength  will  be  greater  than  the  real  maximum  or  breaking 
strength  per  square  inch  of  the  material.  Hence  it  follows 
that  the  3d  and  4th  forms  of  specimen  do  not  indicate  cor- 
rectly the  quality  of  the  material,  furnishing,  as  they  do, 
erroneous  values  for  both  breaking-strength  and  ductility. 


CAST-IRON.  355 


The  quantities  sought  in  such  tests  as  those  described 
above  (with  specimens  of  the  1st,  2d  or  5th  forms)  are,  as 
already  stated  : 

i°.  The  breaking-strength  per  square  inch  of  the  material; 

2°.  The  limit  of  elasticity  of  the  material ; 

3°.  The  yield-point  or  stretch-limit  of  the  material; 

4°.  The  ultimate  contraction  of  area  per  cent : 

5°.  ^he  ultimate  elongation  per  cent  in  a  given  gauged 
length ; 

6°.  The  modulus  of  elasticity. 

The  first  gives,  of  course,  the  tensile  str  :h  of  the  ma- 
terial ;  the  second  and  third  ought  both  to  be  determined,  but 
many  content  themselves  with  the  third  alone,  since  it  is  much 
easier  to  obtain.  While  they  are  commonly  not  far  apart,  it 
is  a  fact  that  certain  kinds  of  stress  to  which  the  piece  may  be 
subjected  may  cause  them  to  become  very  different  from  each 
other.  The  fourth  and  fifth  are  the  usual  ways  of  measuring 
the  ductility  of  the  metal ;  and  while  the  fourth  is  the  most 
definite,  the  fifth  is  very  much  employed,  and  finds  favor  with 
most  iron  and  steel  manufacturers.  The  sixth  is  not  often 
determined  for  commercial  work,  but  it  is  one  of  the  important 
properties  of  the  metal. 

Of  these  six  properties  the  two  most  universally  insisted 
upon  in  specifications  for  material  to  be  used  in  the  construc- 
tion of  structures  or  of  machines  are  ductility,  which  is 
universally  recognized  as  an  all-important  matter,  and  a  suit- 
able breaking-strength  per  square  inch,  both  a  lower  and  an 
upper  limit  being  generally  prescribed  for  this  last. 

On  the  other  hand,  although  cast-iron  and  hard  steel  are 
brittle  metals  when  compared  with  wrought-iron  and  mild 
steel,  nevertheless  it  is  true  that  the  third  and  fourth  forms 
of  specimen  will  show  too  high  results  for  tensile  strength  even 
in  these  materials  on  account  of  the  interference  with  the 
stretch  of  the  metal. 


APPLIED    MECHANICS. 


§  217.  Cast-  Iron.  —  Cast-iron  is  a  combination  of  iron  with 
carbon,  the  most  usual  quantity  being  from  3  to  4  per  cent.  The 
large  amount  of  carbon  which  it  contains  is  its  distinguishing 
feature,  and  determines  its  behavior  in  most  respects.  Besides 
carbon,  cast-iron  contains  such  substances  as  silicon,  phosphorus, 
sulphur,  manganese,  and  others.  A  considerable  amount  (more 
than  1.37  per  cent  as  stated  by  Prof.  Howe)  of  silicon  forces 
carbon  out  of  v  combination  and  into  the  graphitic  form,  thus 
lowering  the  strength. 

Pig-Iron  is  the  result  of  the  first  smelting,  being  obtained 
directly  from  the  blast-furnace.  The  ore  and  fuel  (usually 
coke,  though  anthracite  coal  is  used  to  some  extent,  and  some- 
times charcoal)  are  put  into  the  furnace,  together  with  a  flux, 
which  is  usually  limestone,  in  suitable  proportions.  The  mass 
is  brought  to  a  high  heat,  a  strong  blast  of  heated  air  being  intro- 
duced. The  mass  is  thus  melted,  the  fluid  metal  settling  to 
the  bottom,  while  slag,  which  is  the  result  of  the  combination 
of  the  flux  with  impurities  of  the  ore  and  fuel,  rises  to  the  top. 
The  iron  is  drawn  off  in  the  liquid  state  and  run  into  moulds, 
the  result  being  pig-iron. 

The  metal  usually  contains  from  3  to  4  per  cent  of  carbon, 
a  part  being  chemically  combined  with  the  iron,  and  a  part  in 
the  form  of  graphite.  The  larger  the  proportion  of  combined 
carbon,  the  whiter  the  fracture,  and  the  harder  and  more  brittle 
the  product,  while  the  larger  the  proportion  of  graphite,  the  darker 
the  fracture,  and  the  softer  and  less  brittle  the  product.  That 
which  has  most  of  its  carbon  in  combination  is  called  white  iron, 
while  that  which  contains  a  large  proportion  of  graphite  is  called 
gray  cast-iron. 

Pig-iron  also  contains  silicon,  sulphur,  phosphorus,  etc. 
The  quantity  of  the  first  two  can,  to  a  certain  extent,  be  controlled 
in  the  furnace,  but  not  that  of  the  last,  so  that  if  low  phosphorus  is 
desired,  the  ore  and  the  fuel  used  must  both  be  low  in  phosphorus. 

Gray  cast-iron  has  been,  and  is  sometimes  classified  in  various 


CAST-IRON.  357 


ways,  according  to  the  proportions  of  the  combined  carbon,  and 
of  the  graphite,  but  the  most  modern  practice  is  to  sell,  buy,  and 
specify  the  iron  by  means  of  its  chemical  composition,  and  not  by 
brands. 

That  which  contains  the  largest  amount  of  carbon  in  mechan- 
ical mixture  is,  as  a  rule,  soft  and  fusible,  and  hence  suitable  for 
making  castings  where  precision  of  form  is  the  chief  desidera- 
tum, as  its  fusibility  causes  it  to  fill  the  mould  well.  For  general 
use  in  construction,  where  strength  and  toughness  are  all-import- 
ant considerations,  those  grades  are  required  which  are  neither 
extremely  soft  nor  extremely  hard. 

As  to  the  adaptability  of  cast-iron  to  construction,  it  presents 
certain  advantages  and  certain  disadvantages.  It  is  the  cheapest 
form  of  iron.  It  is  easy  to  give  it  any  desired  form.  It  resists 
oxidation  better  than  either  wrought-iron  or  steel.  Its  com- 
pressive  strength  is  comparatively  high  when  the  castings  are 
small  and  perfect.  On  the  other  hand,  its  tensile  strength  is 
much  less  than  that  of  wrought-iron,  or  that  of  steel,  averaging 
in  common  varieties  from  16000  or  17000  to  about  26000 
pounds  per  square  inch.  It  cannot  be  riveted  or  welded.  It 
is  a  brittle  and  not  a  ductile  material,  it  does  not  give  much 
warning  before  fracture,  and,  while  the  stretch  under  any 
given  load  per  square  inch  is  decidedly  larger  than  that  of 
wrought-iron  or  steel,  its  total  stretch  before  fracture  is  small 
when  compared  with  wrought  iron  and  steel.  One  of  the  dif- 
ficulties in  the  use  of  cast -iron  in  construction  is  its  liability  to 
initial  strains  from  inequality  in  cooling.  Thus  if  one  part  of 
the  casting  is  very  thin  and  another  very  thick,  the  thin  part 
cools  first,  and  the  other  parts,  in  cooling  afterwards,  cause 
stresses  in  the  thin  part. 

The  fracture  of  good  cast-iron  should  be  of  a  bluish-gray 
color  and  close-grained  texture. 

At  one  time  cast-iron  was  extensively  used  for  all  sorts 
of  structural  work,  but  it  was  soon  superseded  by  wrought-iron, 
and  later  by  steel. 

Thus  it   is  no   longer  used   in   bridgework,   nor    for  floor- 


APPLIED    MECHANICS. 


beams  of  a  building,  though  it  is  still  used  to  a  considerable 
extent  for  the  columns  of  buildings;  and  for  this  purpose  it 
has  in  its  favor  the  fact  that  it  resists  the  action  of  a  fire  better 
than  wrought  iron  or  steel.  Thus,  in  the  present  day,  when 
the  steel  skeleton  construction  of  buildings  is  so  extensively 
employed,  it  is  very  necaesary  to  protect  the  steel  beams  and 
columns  by  covering  them  with  some  non-conducting  material, 
as,  otherwise,  they  would  be  liable  to  collapse  in  case  of  fire. 

It  is  used  in  cases  where  the  form  of  the  piece  is  of  more 
importance  than  strength,  and  also  where,  on  account  of  its 
form,  it  would  be  difficult  or  expensive  to  forge  ;  thus  hangers, 
pulleys,  gear-wheels,  and  various  other  parts  of  machinery  of  a 
similar  character  are  usually  made  of  cast-iron,  as  well  as  a 
great  many  other  pieces  used  in  construction.  It  is  also  used 
where  mass  and  hence  weight  is  an  important  consideration, 
as  in  the  bed-plates  and  the  frames  of  machines,  etc. 

Malleable  Iron.  —  When  a  casting,  in  which  toughness  is 
required  is  to  be  made  of  a  rather  intricate  form,  it  is  frequently 
the  .custom  to  malleableize  the  cast-iron,  i.e.,  to  remove  a  part 
of  its  carbon,  and  the  result  is  —  provided  the  casting  is  small 
—  a  product  that  can  be  hammered  into  any  desired  shape  wher* 
c  old,  but  is  brittle  when  hot. 

A  list  of  references  to  some  of  the  principal  experimental 
works  on  the  strength  and  elasticity  of  cast  iron  will  be  given. 

i°.   Eaton  Hodgkinson  :      (a)  Report  of  the  Commissioners  on  the 
Application  of  Iron  to  Railway  Structures. 

(b)  London  Philosophical  Transactions.      1840. 

(c)  Experimental  Researches  on  the  Strength  and  other  Prop- 
erties of  Cast-Iron.      1846. 

2°.  W.  H.  Barlow  :  Barlow's  Strength  of  Materials. 

3°.  Sir  William  Fairbairn  :  On  the  Application  of  Cast  and  Wrought 

Iron  to  Building  Purposes. 
4°.  Major  Wade  (U.S.A.)  :    Report  of  the  Ordnance  Department 

on  the  Experiments  on  Metals  for  Cannon.      1856. 
5°.  Capt.  T.  J.  Rodman  :   Experiments  on  Metals  for  Cannon. 
6°.  Col.  Rosset:  Resistenzadei  Principal!  Metalli  daBocchidi  Fuoco. 


TENSILE  STRENGTH    OF  CAST-IRON.  359 

7°.  Tests  of  Metals  made  on  the  Government  Testing  Machine  at 
Watertown  Arsenal,  1887,  1888,  1889,  1890,  1891,  1892, 
1893,  l894»  1896,  1897,  1898. 

8°.   Transactions  Am.  Soc.  Mechl.  Engrs.  for  1889,  p.  187  et  seq. 

9°.  W.  J.  Keep:  (a)  Transverse  Strength  of  Cast-iron.  Trans.  Am, 
Soc.  Mechl.  Engrs.,  1893. 

(b)  Relative  Tests  of  Cast-iron.     Trans.  Am.  Soc. 

Mechl.  Engrs.,  1895. 

(c)  Transverse  Strength  of  Cast-iron.    Trans.  Am. 

Soc.  Mechl.  Engrs.,  1895. 

(d)  Keep's   Cooling   Curves.       Trans.    Am.    Soc. 

Mechl.  Engrs.,  1895. 

(e)  Strength    of    Cast-iron.      Trans.     Am.     Soc. 

Mechl.  Engrs.,  1896, 
10°.   Bauschinger:  Mittheilungenausdem  Mech.  Tech.  Lab.  Miinchen. 

Heft  12,  1885;  Heft  15,  1887;  Heft  27,  1902;  Heft  28.  1902. 
ii°.   Tetmajer;    Mittheilungen    der    Materialpriifungsanstalt    Zurich. 

Heft  3,  1886;  Heft  4,  1890;   Hefte  5  and  9,  1896. 
12°.   Technology  Quarterly.      October  1888,  page  12  et  seq. 
13°.   Technology  Quarterly.     Vol.  7,  No.  2;  VoL  10.  No.  3. 
14°.  Transactions  of  the  American  Foundrymen's  Association. 
15°.   Transactions  of  the  American  Society  for  Testing  Materials. 

§  218.  Tensile  Strength  of  Cast-iron. — As  the  use  of 
cast-iron  to  resist  tension  has  been  almost  entirely  superseded  by 
that  of  wrought-iron  and  steel,  results  of  tests  of  full-size  pieces 
of  cast-iron  in  tension  are  not  available.  Tensile  tests,  however, 
have  been  extensively  employed  to  determine  the  quality ;  especially 
so  when  cast-iron  cannon  were  in  use;  and  tensile  tests  of  cast- 
iron  are  still  made,  to  a  certain  extent,  for  the  determination 
of  quality.  For  such  tests  standard  specimens  should  be  used, 
and  attempts  are  being  made  to  reduce  their  number. 

As  the  strength  that  should  be  attained  in  such  specimens 
will  become  evident  from  the  Standard  Specifications  of  the 
Am.  Soc.  for  Testing  Materials,  on  page  385  et  seq.,  only  a  few 
tensile  tests  will  be  quoted  here,  and  those,  for  the  purpose  of 


360 


APPLIED   MECHANICS. 


acquainting  the  reader  with  the  results  of  some  tensile  tests  of 
cast-iron. 

About  1840  Eaton  Hodgkinson  made  a  few  experiments  to 
determine  the  laws  of  extension  of  cast-iron,  and  for  this  purpose 
used  rods  lofeet  long  and  i  square  inch  in  section.  The  tables 
of  average  results  are  given  below. 

These  tables  show  that  the  ratio  of  the  stress  to  the  strain  of 
cast-iron  varies  with  the  load,  growing  gradually  smaller  as  the 
load  increases,  that  with  moderate  loads  the  ratio  of  stress  to 

RESULTS  OF  NINE  TENSILE  TESTS.      RESULTS  OF  EIGHT  COMPRESSIVE  TESTS. 


Weights 

Strains  in 

Ratio  of 

Weights 

Strains  in 

Ratio  of 

Laid  on 

Fractions  of 

Stress  to 

Laid  on 

Fractions  of 

Stress  to 

in  Pounds. 

the  Length. 

Total  Strain. 

in  Pounds. 

the  Length. 

Total  Strain. 

1053.77 

O.OOCO7 

14050320 

2064.75 

ocoo.  i  6 

13214400 

1580.65 

o  .  ooo  1  1 

13815720 

4129.49 

0.00032 

12778200 

2107.54 

o  .  ooo  i  6 

13597080 

6194.24 

0.00050 

12434040 

3161.31 

0.00024 

13218000 

8258.98 

0.00066 

12578760 

4215.08 

0.00033 

12936360 

10323.73 

o  .  00083 

12458280 

5268.85 

o  .  00042 

12645240 

12388.48 

O.OOIOO 

12357600 

6322.62 

0.00051 

12377040 

14453.22 

0.00188 

12245880 

7376.39 

0.00061 

12059520 

16517.97 

0.00136 

12132240 

8430.16 

0.00072 

11776680 

18582.71 

0.00154 

12050400 

9483  .  94 

0.00083 

11437920 

20647.46 

0.00172 

12013680 

I0537-7I 

0.00095 

11314440 

24776.95 

0.00208 

11911560 

11591.48 

0.00107 

10841640 

28906.45 

0.00247 

11679720 

12645.25 

O.OOI2I 

10479480 

33030  .  80 

0.00295 

11215560 

13699.83 

0.00139 

9855960 

14793.10 

O.OOI55 

9549120 

strain  for  tension  of  cast-iron  does  not  differ  materially  from 
that  for  compression,  and  that  the  difference  increases  as  the 
load  becomes  greater.  The  agreement  is  even  closer  in  the 
case  of  wrought-iron  and  steel. 

The  gradual  decrease  of  the  ratio  of  stress  to  strain  with  the 
increase  of  load  shows  that  Hooke's  law,  "  Ut  tensio  sic  vis" 
(the  stress  is  proportional  to  the  strain),  does  not  hold  true  in 


RESULTS   OF   TESTS.  361 

cast-iron.  Hence,  strictly  speaking,  cast-iron  has  no  elastic 
limit  and  no  modulus  of  elasticity,  nevertheless  we  are  accustomed 
to  call  the  ratio  of  the  stress  to  the  strain  under  moderate  loads 
the  modulus  of  elasticity  of  the  cast-iron. 

In  making  specifications  intended  to  secure  a  good  quality 
of  'cast-iron  it  is  very  common  to  call  for  a  transverse  test. 
Indeed  the  resolutions  of  the  international  conferences  relative 
to  uniform  methods  of  testing  recommend,  in  the  case  of  cast- 
iron: 

(a)  Test-pieces  to  be  of  the  shape  of  prismatic  bars  no  cm. 
standard  length   (43")   and  to  have  a  section  of  3  cm.  square 
(i".i8),  one  having  an  addition  on  one  end,  from  which  cubes 
can  be  cut  for  compression  tests. 

(b)  Three  such  specimens  to  be  tested  for  transverse  strength. 

(c)  The  tensile  strength  to  be  determined  from  turned  test- 
pieces  20  mm.  (o".785)  diameter  and  200  mm.  (7". 85)  long,  cut 
from  the  two  ends  of  the  test-pieces  broken  by  flexure. 

(d)  The  compressive  strength  to  be  determined  from  cubes 
3  cm.  (i".i8)  on  a  side  cut  from  the  first  specimens,  pressure 
to  be  applied  in  the  direction  of  the  axis  of  the  original  bar. 

These  requirements,  while  calling  for  transverse  tests,  call 
also  for  tensile  and  compressive  tests. 

T  .*  Siandard  Specifications  of  the  Am.  Soc.  for  Testing 
Mater.;.  Js  w.ll  be  found  on  page  385  et  seq. 

Inasmuch  as  the  tensile  strength  has  been,  and  is  also  made 
the  basis  of  specifications  for  cast-iron,  it  is  important  to  con- 
sider what  should  be  attained  in  this  regard. 

For  this  purpose  a  few  tables  of  comparatively  modern  tests 
will  be  given  here,  and  it  will  be  seen  that  in  the  ordinary 
varieties  of  cast-iron  it  is  easy  to  secure  tensile  strengths 
from  16,000  to  25,000  pounds  per  square  inch,  and  that 
more  can  be  secured  by  taking  proper  precautions  in  the 
manufacture. 

Indeed  cast-iron  which,  when  tested  in  the  form  of  a 
grooved  specimen,  shows  a  tensile  strength  of  at  least  30,000 


562 


APPLIED   MECHANICS. 


pounds  per  square  inch  is  called  gun-iron,  this  having  been  a 
requirement  of  the  United  States  Government,  in  the  days  of 
cast-iron  cannon,  for  all  cast-iron  that  was  to  be  used  in  their 
manufacture. 

The  following  table  is  taken  from  a  paper  on  the  Strength 
of  Cast-Iron,  by  Mr.  W.  J.  Keep,  published  in  the  Transactions 
of  the  American  Society  of  Mechanical  Engineers  for  1896, 
and  it  gives  the  averages  of  the  tensile  strengths  of  the  fifteen 
different  series  of  tests  recorded  in  the  paper.  This  table  is 
given  here  merely  as  an  example  of  the  results  that  can  be 
obtained  by  tension  tests  upon  usual  varieties  of  cast-iron. 
The  table  is  as  follows : 


AVERAGES   OF  TENSION    TESTS    OF   ROUND    BARS. 


Area  of  Section, 

Area  of  Section 

Area  of  Section 

Area  of  Section, 

0.375  Sq.  In. 

1.12  Sq    In. 

o  375  Sq.  In. 

1.  12  Sq.  In. 

. 

No.  of 

Breaking  Load 
per  Sq.  Inch. 

Breaking  Load 
per  Sq.  Inch 

Series. 

Breaking  Load 
per  Sq    Inch. 

Breaking  Load 
per  Sq.  Inch. 

J 

2OOOO 

1  57OO 

14800 

2 

20580 

22500 

IO 

oeo^O 

2O4CO 

1  1 

I7OOO 

4 

21850 

19350 

12 

17700 

17500 

5 

22425 

19750 

13 

I4OOO 

2I3OO 

6 

25550 

17200 

H 

24400 

2O3OO 

7 

18950 

17700 

15 

23525 

20500 

8 

17700 

15350 

The  following  table  of  results  of  tension  tests  of  ordinary 
cast-iron  from  another  source  will  also  be  given  for  the  same 
purpose  as  Mr.  Keep's  results : 


CAST-IRON. 


363 


CAST-IRON    TENSION. 


a 

.2 

|.S 

c 

.2 

•O.S 

u 

1-3  "" 

u 

»J  <" 

Dimensions. 

*3 

ll 

Modulus  of 
Elasticity. 

Dimensions. 

c 

S  v 
3  a 
8  en 

Modulus  of 
Elasticity. 

.2  .Q 

f" 

¥ 

Ir 

3d 

.03  X  .04 

.06 

19340 

14857000 

.00  X  .00 

.00 

17100 

13333000 

.03  X  .02 

.05 

23910 

15481000 

.00  X  .02 

.02 

19068 

13680000 

.00  X  98 

.98 

21180 

15238000 

.00  X  .00 

.00 

1  8000 

13333000 

.00  X  97 

•97 

23227 

15881000 

.00  X  .02 

.02 

19299 

12057000 

.ot  X  .06 

.08 

19830 

14539000 

.06  X  .98 

•o? 

17488 

13249000 

.••  X  .03 

•03 

20413 

17632000 

.00  X  .98 

.98 

19500 

13250000 

.93  X  .00 

•93 

16774 

14337000 

.02  X   .02 

•03 

20747 

14543000 

.00  X  -oo 

.00 

18600 

15383000 

.03  X  .03 

.06 

18620 

13434000 

.00  X  .00 

.00 

18000 

16666000 

.00  X  .00 

.00 

18910 

13043000 

.00  X  .00 

.00 

19400 

17911000 

.00  X  .00 

.00 

20950 

15789000 

.00  X  .00 

.00 

19900 

15000000 

.00  X  .00 

.00 

22900 

15000000 

.00  X   .02 

.02 

19594 

13373000 

.00  X  .00 

.00 

22400 

15564000 

.01  x  .03 

.04 

16341 

13108000 

.00  X  .00 

.00 

21300 

15384000 

.01  X  .03 

.04 

'3844 

13640000 

.00  X  .02 

.02 

19692 

i  5966000 

.02  X   .08 

.01 

13798 

11840000 

.ot  X  .03 

.05 

21005 

15075000 

.00  X   .C2 

.02 

17647 

12787000 

08  •<   2; 

•33 

20600 

11900000 

.03  X  .03 

.06 

14025 

12^68000 

.05  X    .03 

•03 

17067 

12676000 

.04  X   .02 

.06 

15083 

13466000 

00  X    .03 

.03 

19900 

12929000 

.02  X  .04 

06 

16874 

9751900 

.08  X  .03 

.02 

16404 

12577000 

.00  X  .00 

.00 

aoooo 

13043000 

i  .co  X  .02 

.02 

16450 

12570000 

Colonel  Rosset,  of  the  Arsenal  at  Turin,  made  a  series  of 
experiments  upon  the  influence  of  the  shape  of  the  specimen 
upon  the  tensile  strength.  For  this  purpose  he  used  specimens 
with  shoulders ;  and,  among  other  tests,  he  compared  the 
strength  of  the  same  iron  by  using  specimens  the  lengths  of 
whose  smallest  parts  were  respectively  i  metre,  30  millimetres, 
and  o  millimetres,  with  the  following  results  :  — 


Length  of  Specimen. 

Tensile  Strength,  in  Ibs.,  per  Square  Inch. 

ist  Cannon. 

2d  Cannon. 

3d  Cannon. 

i  metre      .     . 
30  millimetres  . 
o  millimetres  . 

31291 
3257I 
33993 

25601 

34562 
36411 

28019 
30011 
30011 

APPLIED   MECHANICS. 


It  will  thus  be  seen  that,  before  we  can  decide  upon  the 
quality  of  cast-iron  as  affected  by  the  tensile  strength,  it  is 
necessary  to  know  the  length  of  that  part  of  the  specimen 
which  has  the  smallest  area.  Colonel  Rosset's  tests  of  cast 
iron  were  almost  entirely  confined  to  high-grade  irons,  suitable 
to  use  in  cannons. 

He  deduced,  for  mean  value  of  the  modulus  of  elasticity  of 
the  specimens  i  metre  in  length,  20419658  Ibs.  per  square  inch  : 
this,  of  course,  is  a  modulus  only  adapted  to  these  high  grades, 
and  is  not  applicable  to  common  cast-iron. 

§  219.  Cast-Iron  Columns.  —  In  consequence  of  the  high 
compressive  strength  shown  by  cast-iron  when  tested  in  small 
pieces,  and  in  pieces  free  from  imperfections,  it  was  once 
considered  a  very  suitable  material  for  all  kinds  of  columns. 
Nevertheless,  its  use  for  the  compression  members  of  bridge 
and  roof  trusses  has  been  abandoned;  cast-iron  having  been 
displaced  first  by  wrought-iron  and  subsequently  by  steel, 
which  is  the  substance  now  in  use  for  these  purposes. 

The  principal  reasons  for  the  change  are  the  lack  of 
ductility,  and  the  consequent  brittleness  of  cast-iron,  that  it 
cannot  be  riveted,  and  that  if  it  breaks  it  cannot  be  eas'ly 
repaired.  Cast-iron  is,  however,  used  to  a  very  considerable 
extent  for  the  columns  of  buildings. 

The  Gordon,  the  so-called  Euler,  and  the  Hodgkinson 
formulae  for  the  breaking-strength  of  cast-iron  columns,  have 
all  been  given  in  paragraphs  208,  2080,  and  209.  They  are, 
however,  all  based  upon  tests  made  upon  very  small  columns, 
and  do  not  give  results  agreeing  with  the  tests  of  such  full-size 
columns  as  are  used  in  practice.  We  will  next  consider, 
therefore,  the  tests  that  have  been  made  upon  full-size  cast-iron 
columns,  and  the  conclusions  that  are  warranted  in  the  light  of 
these  tests. 

Two  sets  of  tests  of  cast-iron  mill  columns  have  been  made 
on  the  Government  testing-machine  at  Watertown  Arsenal;  an 
account  of  these  sets  of  tests  is  published  in  their  reports  of 
1887  and  of  1888. 


CAST-IRON   COLUMNS.  365 


The  first  lot  consisted  of  eleven  old  cast-iron  columns,  which 
had  been  removed  from  the  Pacific  Mills  at  Lawrence,  Mass., 
during  repairs  and  alterations. 

The  second  lot  consisted  of  five  new  cast-iron  columns  cast 
along  with  a  lot  that  was  to  be  used  in  a  new  mill. 

Of  these  five,  the  strength  of  two  was  greater  than  the 
capacity  of  the  testing-machine,  hence  only  three  were  broken  ; 
while  in  the  case  of  the  other  two  the  test  was  discontinued 
when  a  load  of  800000  Ibs.  was  reached.  All  the  columns  con- 
tained a  good  deal  of  spongy  metal,  which  of  course  rendered 
their  strength  less  than  it  would  otherwise  have  been  ;  never- 
theless, inasmuch  as  this  is  just  what  is  met  with  in  building, 
it  is  believed  that  these  tests  furnish  reliable  information  as  to 
what  we  should  expect  in  practice,  and  that  this  information 
is  much  more  reliable  than  any  that  can  be  derived  from  test- 
ing small  columns. 

In  all  the  tests  the  compressions  were  measured  under  a 
large  number  of  loads  less  than  the  ultimate  strength ;  but  in- 
asmuch as  it  is  not  possible,  in  the  case  of  cast-iron,  to  fix  any 
limits  within  which  the  stress  is  proportional  to  the  strain,  no 
attempt  will  here  be  made  to  compute  the  modulus  of  elas- 
ticity. Hence  there  will  be  given  here  a  table  showing  the 
dimensions  of  the  columns  tested,  their  ultimate  strengths, 
and,  in  those  cases  where  they  were  measured,  the  horizontal 
and  vertical  components  of  their  deflections,  measured  at  the 
time  when  their  ultimate  strengths  were  reached,  as  the  Govern- 
ment machine  is  a  horizontal  machine.  A  glance  at  the  table 
will  make  it  evident  that  we  Cannot,  in  the  case  of  such  columns, 
rely  upon  a  crushing  strength  any  greater  than  25000  or  30000 
Ibs.  per  square  inch  of  area  of  section.  Hence  it  would  seem 
to  the  writer  that,  in  order  to  proportion  a  cast-iron  column 
to  bear  a  certain  load  in  a  building,  we  should  determine  the 
outside  diameter  in  such  a  way  as  to  avoid  an  excessive  ratio 
of  length  to  diameter ;  if  this  ratio  is  not  much  in  excess 


APPLIED    MECHANICS. 


of  twenty,  the  extra  stress  produced  by  any  eccentricity  of  the 
load  due  to  the  deflection  of  the  column  will  be  very  slight. 
At  the  same  time  see  that  the  thickness  of  metal  is  sufficient 
to  insure  a  good  sound  casting. 

Now,  having  figured  the  column  in  this  way,  compute  the 
outside  fibre  stress  (using  the  method  of  §  207)  that  would 
occur  with  the  loading  of  the  floors  assumed  to  be  such  as  to 
give  as  great  an  eccentricity  as  it  is  possible  to  bring  upon  the 
column.  If  this  distribution  of  the  load  is  one  that  is  likely 
to  occur,  then  the  maximum  fibre  stress  in  the  column  due  to 
it  ought  not  to  be  greatly  in  excess  of  5000  Ibs.  per  square 
inch  ;  but  if  it  is  one  which  there  is  scarcely  a  chance  of  realiz- 
ing, then  the  maximum  fibre  stress  under  it  might  be  allowed 
to  reach  10000  Ibs.  per  square  inch.  If  by  adopting  the  di~ 
mensions  already  chosen  these  results  can  be  obtained,  we  may 
adopt  them  ;  but  if  it  is  necessary  to  increase  the  sectional 
area  in  order  to  accomplish  them,  we  should  increase  it. 

Another  matter  that  should  be  referred  to  here  is  the  fact 
that  a  long  cap  on  a  column  is  more  conducive  to  the  produc- 
tion of  an  eccentric  loading  than  a  short  one  ;  hence,  that  a 
long  cap  is  a  source  of  weakness  in  a  column. 

Other  sources  of  weakness  in  cast-iron  columns  are  spongy 
places  in  the  casting  (which  correspond  in  a  certain  way  with 
knots  in  wood),  and  also  an  inequality  in  the  thickness  of  the 
two  sides  of  the  column,  the  result  of  this  being  the  same  as 
that  of  eccentric  loading;  and  it  is  especially  liable  to  occur  in 
consequence  of  the  fact  that  it  is  the  common  practice  to  cast 
columns  on  their  side,  and  not  on  end.  The  engineer  should, 
however,  inspect  all  columns  to  be  used  in  a  building,  and  reject 
any  that  have  the  thickness  of  the  shell  differing  in  different 
parts  by  more  than  a  very  small  amount. 

A  series  of  tests  of  full-size  cast-iron  columns  was  made  by 
the  Department  of  Buildings  of  New  York  City,  under  the 
direction  of  Mr.  W.  W.  Ewing,  in  December,  1897,  upon  the 


CAST-IKON  COLUMNS. 


367 


Remarks. 

rt  T3       rt            rt  ^aJ 
w  nj     •  w  T3        W 

iple  flexure.  Post  con- 
t  middle  before  testing, 
iple  flexure. 

C 

U 
V 

en 

»-  ^  H  2  |i        o  ,_,  ^     ** 

«j 

ID 

U 

aanx; 

e  flexure 

t 

3 
X 

U,^    *u*         *>1'X      X 

3 

3 
X 

3 
X 

c  g»2  c  §     1  JfS   S 

« 

« 

« 

1C 

oS^o-S    §"S-e-c 

cx 

CX 

CX 

ex 

a. 

a. 

.0^  -5  £.«  w.  _^tc  ^  ^ 

"°   S              "3        T3  -°     -° 

1 

-o 

•a 

3 

'3 

j5' 

3 

'rt 

js 
'3 

*o  2f  'O 
<u       <u 

(3          (3 

i 

'rt 

rt             rt             rt        rt      rt 

'rt 

"rt 

'1 

lid 

3  "5  -2 

« 
o 

° 

0^ 
0 

CO          rfr 

C^              HH 

M              M 

en 

O 

«t  i 

a  s   •  > 

.2  c/5 

-i- 

0 

8 

0 
in 

6 

m       en 

6     6 

m 

oo 

O 

'S  a      s 

Q  2        B 
D 

o 

6 

in 
N 

0 

O       co 
en      oo 

M              6 

CO 

en 

0 

"  -S         r 

rt  tuo       a    . 

5  w     ^ 

O              O              Q         O      O 
co             m            O        "^    r^ 
oo             en            m      co      en 
co             en            en      o      O 
en            ^            en       w     en 

C-O 

cr- 

C) 

0 
co 

CO 

0 

o 

in 

CO 

en 

Cl 

0 

CO 

o 
en 

en 

R     2 

m        r^ 

8 

lip 

p  w 

1      1      1    1  §• 

N              *,              rf       in     w 
M              en             O         M     o 
in              in              ^-        en     en 

352OOO 

in 

° 

in 

0 

1 

i 

CO 

00 

m 

en 

§     8 

O           CM 
•*!•          O 

in       r^* 

m        rt 

o 

CO 

en 

,, 

vn            co              m        en    co 

H 

« 

CO 

o 

8 

0 

Cl 

8 
-f 

0 

4- 

-t 

r^ 

0             Tf 

to        ^ 

t***         QO 

! 

f|f  s 

5  tJ  •<  cr 

i^ 

.be  <i 

'  V     £ 

in     O 
rt            rt            rt       *^    ^ 

eg 

. 

ST 

CN 

o 

m 

ft 

IO         CO 

M 

en 

*     2     S   "  " 

" 

co 

«n 

in 

O 

^            co            en       r^»    en 
r*^            t^«           \o        ^     ^ 

5 

in 

O 

-1- 

CM 

* 

M 

§  g 

1 

11 
U 

O             M             N        en     *f 

in 

8: 

« 

r-. 
$ 

1 

1 

1 

•sutunjoD 

M3JSI 

•sutunjoo 

PIO 

368 


APPLIED     MECHANICS. 


••i 

* 


513 


Mb! 

--tMAJsJi. 


S-'OI- 


Sg-'6-OI 


CAST-IRON    COLUMNS. 


369 


370 


APPLIED    MEGHAN rCS. 


hydraulic  press  of  the  Phoenix  Bridge  Works.  This  press 
weighs  the  load  on  the  specimen  plus  the  friction  of  the  piston, 
the  latter  being,  of  course,  a  variable  quantity.  Nevertheless 
great  pains  were  taken  to  determine  this  friction,  and  hence 
the  results  are  doubtless  substantially  correct. 

The  results  are,  it  will  be  seen,  similar  to  those  obtained  in 
the  Watertown  tests.  The  table  of  results  is  given  below, 
and  no  farther  comments  are  needed.  Subsequently  tests 
were  made  to  determine  the  strength  of  the  brackets.  For 
this,  however,  the  reader  is  referred  to  the  Report  itself,  or  to 
Engineering  News  of  January  2O,  1898,  and  for  further  details 
of  the  tests  of  the  columns,  to  the  Report  itself,  or  to  Engi- 
neering  News  of  January  13,  1898. 


Column 
Number. 

Length, 
Inches. 

Outside 
Diameter, 
Inches. 

Average 
Thick- 
ness, 
Inches. 

Breaking 
Load, 
Lbs. 

Average 
Area  Sec- 
tion, 
Sq.  In. 

Inches 

/ 
P 

Break- 
ing 
Load 
per  sq. 
in.,  IDS. 

I 

190.25 

15 

1356000 

43.98 

4.96 

.38.36 

30832 

II 

190.25 

15 

^ 

1330000 

49-03 

4.92 

38.67 

27126 

B* 

190.25 

15 

\ 

1198000 

49-03 

4.92 

38.67 

24434 

B< 

190.25 

15* 

\ 

1246000 

49.48 

4.98 

38.20 

25l8l 

5 

190.25 

15 

H 

1632000 

50.91 

4.91 

38.75 

32057 

over 

over 

6 

I90.2§ 

15 

'A 

2082000 

51.52 

4,90 

38.73 

**            '  +* 

40411 

XVI 

1  60 

81  to  7f 

i 

651000 

21.99 

2.50 

64.00 

29604 

XVII 

1  60 

8 

'& 

645600 

22.87 

2.48 

64.52 

28229 

7 

t20 

6TV 

*& 

455200 

17.64 

1.78 

67.41 

25805 

8 

1  2O 

6ft 

i*V 

474100 

17-37 

1.8o 

66.67 

27236 

CAST-IRON    COLUMNS. 


50000 
40000 
30000 
20000 


70   80 


90   100   110   120 


Abscissge=  length  di- 
vided by  radius  of 
gyration  of  small- 
est section. 

Ordinates:=  breaking 
strengths  per 
square  inch  of 
smallest  section. 


130     no     150 


CAST-IRON   COLUMNS. 


371 


The  cut  on  page  370  shows  a  graphical  representation  of 
the  preceding  tests  of  full-size  cast-iron  columns. 

In  Heft  VIII  (1896)  of  the  Mitt.  d.  Materialpriifungsanstalt 
in  Zurich  is  an  account  of  296  cast-iron  struts  tested  by  Prof. 
Tetmajer;  46  being  3  cm.  (i".  1 8)  square  will  not  be  men- 
tioned farther.  The  other  250  were  hollow  circular,  the  inside 
diameters  being  10  cm.  (3". 94),  12  cm.  (4". 72),  or  15  cm. 
(5".9i);  the  thicknesses  being  i  cm.  (o".39)  oro.8  cm.  (0^.31). 
The  lengths  varied  from  4  m.  (13'.  12)  to  20  cm.  (?".g).  They 
are  not  the  most  usual  thicknesses  of  columns  for  buildings, 
though  used  to  a  considerable  extent.  They  might  be  called 
cast-iron  pipe  columns.  The  following  table  contains  all  those 
250  cm.  (8'.2)  long  and  over,  and  i  cm.  thick,  and  one  set  of 
those  0.8  cm.  thick.  This  will  exhibit  the  character  of  the  results 

for  such  columns  of  usual  lengths.    In  computing  —  the  actual 


Thickness  o".  39. 

Thickness  o".3i. 

Outside 

Ultimate 

Outside 

Ultimate 

No.  of 
Test. 

Length, 
Feet. 

Diame- 
ter, 

I 

p 

Strength, 
Pounds 

No.  of 
Test. 

Length, 
Feet. 

Diame- 
ter, 

/ 
p 

Strength, 
Pounds 

Inches. 

per  sq.  in. 

Inches, 

per  sq.  in. 

55 
56 

9.84 
9.84 

S3 

77?:7 

18481 
20761 

207 
208 

13.12 
13.12 

4.62 
.61 

103.9 
103.9 

11518 

11660 

57 

8.20 

4.76 

63.4 

28156 

209 

n.48 

•58 

91.1 

16922 

58 

8.20 

4.78 

63«9 

2986-2 

210 

n.48 

•59 

91.9 

-0577 

69 
70 
7* 

9.84 

9.84 

8.20 

5-63 

5.6z 
5.65 

64.4 
64.4 
53-3 

24174 
32564 
36546 

211 
212 
213 

9.84 
9.84 

8.20 

•56 
.60 
.56 

78.9 
77-7 
65-4 

194.12 
19482 
3*843 

72 

8.20 

5.63 

53-4 

47353 

3I4 

8.20 

4.61 

64.7 

33  '33 

86 

9.84 

6.69 

53-2 

32564 

225 

13.12 

5-41 

87.8 

15216 

87 

9.84 

6.67 

53-3 

34270 

226 

13-12 

5-43 

87.5 

17623 

88 

8.20 

6-73 

43-9 

44224 

227 

11.48 

5-41 

76.7 

22326 

89 

8.20 

6.69 

44.1 

46642 

228 
229 

11.48 
9.84 

5-39 
5-39 

77-3 
66.4 

2I3I» 

23748 

230 

9.84 

5-41 

66.1 

23463 

231 

8.20 

5-41 

54-  8 

38110 

232 

8.20 

5-41 

54-5 

36688 

243 

13.12 

6.56 

71.8 

22041 

244 

13.12 

6-54 

71.9 

24885 

245 

11.48 

6.56 

62.6 

27729 

246 

11.48 

6.56 

62.5 

28156 

247 

9.84 

6-53 

S3-9 

35520 

248 

9.84 

6-54 

53-9 

31853 

249 

8.20 

6.56 

44-8 

4*949 

250 

8.20 

6.56 

44.8 

453^2 

372  APPLIED    MECHANICS. 

length  of  the  strut  has  been  used,  whereas  Tetmajer  adds  to 
this  9 ".84,  the  thickness  of  the  platforms  of  the  machines,  as 
they  bore  o;i  knife-edges. 

Prof.  Bauschinger  of  Munich  made  two  series  of  tests  of 
full-size  cast- and  of  wrought-iron  columns  to  determine  the 
effect  of  heating  them  red-hot  and  sprinkling  them  with  water 
while  under  load.  They  were  loaded  in  his  testing-machine 
with  their  estimated  safe  load  as  calculated  from  the  formulas. 

For  cast-iron, 


19912^4 

./" 

For  wrought-iron, 


i  -|-  0.0006-5 


11378^4 
*'  r, 

i  -f-  0.00009  2 

where  P  =  safe  load  (factor  of  safety  five),  A  =  area  of  section, 
/  =  length,  p  —  least  radius  of  gyration,  pounds  and  inches 
being  the  units. 

A  fire  was  made  in  a  U-shaped  receptacle  under  the  post, 
so  arranged  that  the  flames  enveloped  the  post.  The  tem- 
perature was  determined  from  time  to  time  by  means  of  alloys 
of  different  melting-points  ;  and  the  horizontal  and  vertical 
components  of  the  deflections  were  read  off  on  a  dial  as  indi- 
cated by  a  hand  attached  to  the  post  by  a  long  wire.  The 
post  was  also  examined  for  cracks  or  fractures. 

In  the  1884  series  he  tested  six  cast-iron  posts  of  various 
styles,  and  three  wrought-iron  posts,  one  of  them  being  made 
of  channel-irons  and  plates  put  together  with  screw-bolts,  one 
of  I  irons  and  plates  also  put  together  with  screw-bolts,  and 
one  hollow  circular. 

The  details  of  the  tests  will  not  be  given  here,  but  only 
Bauschinger's  conclusions.  He  said  : 

That  wrought-iron  columns,  even  under  the  most  favorable 


CAST- 1  RON-   COLUMNS.  373 

adjustment  of  their  ends  and  of  the  manner  of  loading,  bend 
so  much  that  they  cannot  hold  their  load,  sometimes  with  a 
temperature  less  than  600°  Centigrade,  and  always  when  they 
are  at  a  red  heat ;  and  this  bending  is  accelerated  by  sprink- 
ling on  the  opposite  side,  even  when  only  the  ends  of  the  post 
are  sprinkled. 

.  That  under  similar  circumstances  cast-iron  posts  bend,  and 
this  bending  is  increased  by  sprinkling  ;  but  it  does  not  exceed 
certain  limits,  even  when  the  post  is  red  for  its  entire  length 
and  the  stream  of  water  is  directed  against  the  middle,  and  the 
post  does  not  cease  to  bear  its  load  even  when  cracks  are  de- 
veloped by  the  sprinkling.  Only  when  both  ends  of  a  cast-iron 
post  are  free  to  change  their  directions  does  sprinkling  them 
at  the  middle  of  the  opposite  side  when  they  are  red  make 
them  break,  but  such  an  unfavorable  case  of  fastening  the  ends 
hardly  ever  occurs  in  practice. 

That  the  cracks  in  the  columns  tested  occurred  in  the 
smooth  parts,  and  not  at  corners  or  projections. 

That  the  result  of  these  tests  warns  us  to  be  much  more 
prudent  in  regard  to  the  use  of  wrought-iron  in  building.  If 
posts  which  are  subjected  to  a  longitudinal  pressure  bend  so 
badly  when  subjected  to  heat  on  one  side  that  they  lose  the 
power  of  bearing  their  load,  how  much  more  must  this  be  the 
case  with  wrought-iron  beams  ;  and  he  urges  the  importance  of 
making  more  experiments. 

In  Heft  XV  of  the  Mittheilungen  he  says  that  the  results 
were  criticised  in  two  ways,  viz.  :  Moller  claiming  that  he 
should  have  used  different  constants,  and  Gerber  that  the 
wrought-iron  posts  were  not  properly  made. 

Bauscliinger  therefore  concluded  to  make  a  new  set  of  tests, 
and  for  this  purpose  he  had  made  two  cast-iron  and  five 
wrought-iron  columns — the  former  being  carefully  cast,  but  on 
the  side,  while  the  wrought-iron  ones  were  made  by  a  bridge 
company  of  very  good  reputation,  and  four  of  them  were 
similar  to  those  made  at  the  time  for  a  new  warehouse  in 
Hamburg. 


374  APPLIED    MECHANICS. 


The  tests  were  made  just  as  before,  and  the  following  are 
his  conclusions : 

That  when  wrought-iron  posts  are  as  well  constructed  as 
the  two  referred  to,  they  resist  fire  and  sprinkling  tolerably 
well,  though  not  as  well  as  cast-iron ;  but  that  posts  con- 
structed like  the  other  three,  even  with  the  fire  alone,  and 
before  the  sprinkling  begins,  get  so  bent  that  they  can  no 
longer  hold  their  load.  Good  construction  requires  that  the 
rows  of  rivets  shall  extend  through  the  entire  length  of  the 
post,  and  the  rivets  should  be  quite  near  each  other ;  but  the 
tests  are  not  extensive  enough  to  show  what  are  the  necessary 
requirements  to  make  wrought-iron  posts  able  to  stand  fire  and 
sprinkling ;  in  order  to  know  this  more  experiments  are  needed. 

In  Dingler's  Polytechnisches  Journal  for  1889,  page  259  et 
seq.,  is  an  article  by  Professor  A.  Martens,  of  Berlin,  uprn 
the  behavior  of  cast-  and  wrought-iron  in  fires,  considering 
especially  the  burning  of  a  large  warehouse  in  Berlin,  and 
advocating  the  protection  of  iron-work  by  covering  it  with 
cement.  He  says  that  there  are  two  series  of  tests  upon 
this  subject,  one  of  which  is  the  tests  of  Bauschinger  already 
explained,  and  the  other  a  set  of  tests  made  by  Moller  and 
Luhmann. 

No  detailed  account  of  these  tests  will  be  given  here,  but 
only  Holler's  conclusions,  as  stated  by  Prof.  Martens,  which 
are  as  follows: 

i°.  With  ten  cast-iron  posts  he  could  not  get  any  cracks 
by  sprinkling  at  a  red  heat;  but  it  is  to  be  noted  that  his  were 
new  posts,  while  those  used  in  Bauschinger's  first  series  were 
old  ones,  and  that  those  in  Bauschinger's  second  series,  which 
were  new  and  very  carefully  cast,  did  not  show  cracks  either. 

2°.  He  claims  that  while  the  cracks  would  allow  the  post 
still  to  bear  a  centre  load,  it  could  not  bear  an  eccentric  load 
or  a  transverse  load. 

3°.  "He  claims  that  the  load  on  a  cast-iron  post  should  be 
limited  to  one  which  shall  not  produce  sufficient  bending  to 
bring  about  a  tensile  stress  anywhere  when  the  post  is  bent  by 
the  heat  and  sprinkling. 


TRANSVERSE  STRENGTH  OF  CAST-IRON.  375 

4°.  He  claims  that  in  either  cast-  or  wrought-iron  posts,  if 
the  ends  are  not  fixed,  the  ratio  of  length  to  diameter  should 
not  exceed  I o,  whereas  if  they  are  it  should  not  exceed  17; 
also,  that  there  is  no  such  thing  as  absolute  safety  from  fire 
with  iron. 

5°.  A  covering  of  cement  delays  the  action  of  the  fire,  and 
that  therefore  such  a  covering  is  a  protection  to  the  post 
against  excessive  one-sided  heating  and  cooling. 

6°.  Cast-iron  is  more  likely  to  have  at  any  one  section  a 
collection  of  hidden  flaws  than  wrought-iron. 

§  220.  Transverse  Strength  of  Cast-iron. — At  one  time 
cast-iron  was  very  largely  used  for  beams  and  girders  in  build- 
ings to  support  a  transverse  load.  Its  use  for  this  purpose  has 
now  been  almost  entirely  abandoned,  as  it  has  been  superseded 
by  wrought-iron  and  steel. 

A  great  many  experiments  have  been  made  on  the  trans- 
verse strength  of  cast-iron ;  the  specimens  used  in  some  cases 
being  small,  and  in  others  large.  The  records  of  a  great  many 
experiments  of  this  kind  are  to  be  found  in  the  first  four  books 
of  the  list  already  enumerated  in  §  217.  The  details  of  these 
tests  will  not  be  considered  here,  but  an  outline  will  be  given 
of  some  of  the  main  difficulties  that  arise  in  applying  the  results 
and  in  using  the  beams. 

Cast-iron  is  treacherous  and  liable  to  hidden  flaws ;  it  is 
brittle.  It  is  also  a  fact  that  in  casting  any  piece  where  the 
thickness  varies  in  different  parts,  the  unequal  cooling  is  liable 
to  establish  initial  strains  in  the  metal,  and  that  therefore 
those  parts  where  such  strains  have  been  established  have 
their  breaking-strength  diminished  in  proportion  to  the  amount 
of  these  strains. 

In  the  case  of  cast-iron  also,  the  ratio  of  the  stress  to  the 
strain  is  not  constant,  even  with  small  loads,  and  is  far  from 
constant  with  larger  loads  ;  also,  inasmuch  as  the  compressive 
strength  is  far  greater  than  the  tensile,  it  follows  that,  in  a 
transversely  loaded  beam  which  is  symmetrical  above  and  be- 
low the  middle,  the  fibres  subjected  to  tension  approach  their 


376  APPLIED    MECHANICS. 

full  tensile  strength  long  before  those  subjected  to  compression 
are  anywhere  near  their  compressive  strength.  The  result  of 
all  this  is,  that  if  a  cast-iron  beam  be  broken  transversely,  and 
the  modulus  of  rupture  be  computed  by  using  the  ordinary 
formula, 

f-My 
7     "   I  ' 

we  shall  find,  as  a  rule,  a  very  considerable  disagreement  be- 
tween the  modulus  of  rupture  so  calculated  and  either  the 
tensile  or  compressive  strength  of  the  same  iron.  Indeed, 
Rankine  used  to  give,  as  the  modulus  of  rupture  for  rectangu- 
lar cast-iron  beams,  40000  Ibs.  per  square  inch,  and  for  open- 
work beams  17000  Ibs.  per  square  inch,  which  latter  is  about 
the  tensile  strength  of  fairly  good  common  cast-iron. 

A  great  deal  has  been  said  and  written,  and  a  good  many 
experiments  have  been  made,  to  explain  this  seeming  disagree- 
ment between  the  modulus  of  rupture  as  thus  computed,  and 
the  tensile  strength  of  the  iron.  Barlow  proposed  a  theory 
based  upon  the  assumption  of  the  existence  of  certain  stresses 
in  addition  to  those  taken  account  of  in  the  ordinary  theory  of 
beams,  but  his  theory  has  no  evidence  in  its  favor. 

Rankine  claimed  that  the  fact  that  the  outer  skin  is  harder 
than  the  rest  of  the  metal  would  serve  to  explain  matters,  but 
this  would  not  explain  the  fact  that  the  discrepancy  exists  in 
the  case  of  planed  specimens  also. 

Neither  Barlow  nor  Rankine  seems  to  have  attempted  to 
find  the  explanation  in  the  fact  that  the  formula 


assumes  the  proportionality  of  the  stress  to  the  strain,  and 
hence  that  is  less  and  less  applicable  the  greater  the  load,  and 
hence  the  nearer  the  load  is  to  the  breaking  load.  An  article 
by  Mr.  Sondericker  in  the  Technology  Quarterly  of  October, 


TRANSVERSE   STRENGTH  OF  CAST-IRON.  377 

1888,  gives  an  account  of  some  experiments  made  by  him  to 
test  the  theory  that  "  the  direct  stress,  tension,  or  compression, 
at  any  point  of  a  given  cross-section  of  a  beam,  is  the  same 
function  of  the  accompanying  strain,  as  in  the  case  of  the  cor- 
responding stress  when  uniformly  distributed,"  and  the  results 
bear  out  the  theory  very  well  ;  hence  it  follows  that,  if  we  use 
the  common  theory  of  beams,  determining  the  stresses  as  such 
multiples  of  the  strains  as  they  show  themselves  to  be  in  direct 
tensile  and  compressive  tests,  the  discrepancies  largely  vanish, 
and  those  that  are  left  can  probably  be  accounted  for  by  initial 
stresses  due  to  unequal  rate  of  cooling,  and  by  the  skin,  or 
by  lack  of  homogeneity.  In  the  same  article  he  quotes  the 
results  of  other  tests  bearing  more  or  less  on  the  matter,  and 
there  will  be  quoted  here  the  table  on  page  378. 
If,  therefore,  we  wish  to  make  use  of  the  formula 


y 

in  calculating  the  strength  of  cast-iron  beams,  we  cannot  use 
one  fixed  value  of  f  for  all  beams  made  of  one  given  quality 
of  cast-iron,  but  we  shall  have  to  use  a  very  varying  modulus 
of  rupture,  varying  especially  with  the  form,  and  also  with  the 
size  of  the  beam  under  consideration.  Now,  in  order  to  do 
this,  and  obtain  reasonably  correct  results,  we  need,  wherever 
possible,  to  use  values  of  f  that  have  been  deduced  from  ex- 
periments upon  pieces  like  those  which  we  are  to  use  in  prac- 
tice, and  under,  as  nearly  as  possible,  like  conditions. 

There  are  not  very  many  records  of  such  experiments  avail- 
able, and,  in  cases  where  we  cannot  obtain  them,  it  will  prob- 
ably be  best  to  use  a  value  of  f  no  greatei  than  the  tensile 
strength  for  complicated  forms,  and  forms  having  thin  webs. 
For  pieces  of  rectangular  or  circular  section  we  might  probably 
use,  for  good  fair  cast-iron,  25000  to  30000  Ibs.  per  square 
inch. 

A  few  tests  of  the  character  referred  to  have  been  made  in 
the  engineering  laboratories  of  the  Massachusetts  Institute  of 


3;8 


APPLIED    MECHANICS. 




Modulus  oil 

Form  of 
Beam    Sec- 

Tensile 
Strength, 

Rupture 

,      My 
J  —  ~7~< 

Ratio. 

Condition 
of 

Experimenter. 

tion, 

\bs.  per  Sq. 
In. 

/ 
Ibs.  perSq. 

Specimen. 

In. 

19850 

41320 

2.08 

Turned 

C.  Bach.* 

x?H^, 

16070 

35500 

2.21 

Turned 

Considere.f 

34420 

63330 

1.84 

Turned 

Considere. 

\Hx 

24770 

54390 

2.19 

Turned 

Robinson  and  Segundo.  J 

25040 

46280 

1.85 

Rough 

Robinson  and  Segundo. 

Mean. 

2.03 

16070 

29250 

1.82 

Planed 

Considere. 

ijlflp 

36270 

58760 

1.62 

Planed 

Considfere. 

19090 

33740 

1-77 

Planed 

C.  Bach. 

Mean. 

1.74 

19470 

34000 

1-75 

Planed 

C.  Bach. 

in 

31430 

49030 

1.56 

Planed 

Considere. 

Hi 

19880 

33860 

1.70 

Planed 

Sondericker. 

24770 

42340 

1.71 

Planed 

Robinson  and  Segundo. 

g^^ 

25040 

42IIO 

1.68 

Rough 

Robinson  and  Segundo. 

Mean. 

1.68 

19470 

28150 

1-45 

Planed 

C.  Bach. 

16070 

225OO 

1.40 

Planed 

Considere. 

31860 

36640 

I-I5 

Planed 

Considere. 

25040 

3I3IO 

1.25 

Rough 

Robinson  and  Segundo, 

Mean. 

I-3I 

16070 

23780 

1.48 

Planed 

Considere. 

n 

31290 

34730 

I.  II 

Planed 

Considere. 

n 

18050 

24550 

1.36 

Planed 

Sondericker. 

^^ 

22470 

26150 

1.16 

Rough 

Burgess  and  Viel6.  § 

Mean. 

1.28 

«  See  Zeitschrift  des  Vereines  Deutscher  Ingenieure,  Mar.  3d  and  loth,  1888. 

t  See  Annales  des  Fonts  et  Chausse"es,  1885. 

t  See  Proceedings  Institute  of  Civil  Engineers,  Vol  86. 

5  Sec  Proceedings  Am.  Soc.  Mecbl.  Engrs.  1889,  pp.  187  et  seq. 


TRANSVERSE  STRENGTH  OF  CAST-IRON. 


379 


Technology,  and  a  brief  statement  of"  them  will  be  given  here. 
The  first  that  will  be  referred  to  here  is  a  series  of  experiments 
made  by  two  students  of  the  Institute,  an  account  of  which  is 
given  in  the  Proceedings  of  the  American  Society  of  Mechani- 
cal Engineers  for  1889,  pp.  187  et  seq. 

The  object  of  this  investigation  was  to  determine  the  trans- 
verse strength  of  cast-iron  in  the  form  of  window  lintels,  and 
also  the  deflections  under  moderate  loads,  and  from  the  latter 
to  deduce  the  modulus  of  elasticity  of  the  cast-iron,  and  to 
compare  it  with  the  modulus  of  elasticity  of  the  same  iron,  as 
determined  from  tensile  experiments  ;  also  the  tensile  strength 
and  limit  of  elasticity  of  specimens  taken  from  different  parts 
of  che  lintel  were  determined. 

The  iron  used  was  of  two  qualities,  marked  P  and  5  respec- 
tively. 

The  tensile  specimens  were  cast  at  the  same  time,  and  from 
the  same  run  as  the  lintels. 

Besides  this,  one  of  each  kind  of  window  lintels  was  cut  up 
into  tensile  specimens,  and  the  specimens  were  so  marked  as  to 
show  from  what  part  of  the  lintel  they  were  cut. 

The  tables  of  tests  will  now  be  given,  and  the  following  ex- 
planation of  the  symbolism  employed. 

P  and  S  are  used,  as  already  stated,  to  denote  the  quality 
of  the  iron. 

A  and  B  are  used  to  denote,  respectively,  that  the  specimen 
was  unplaned  or  planed. 

I,  2,  3,  etc.,  denote  the  number  of  the  test  made  on  that 
particular  kind  and  condition. 


380 


APPLIED    MECHANICS. 


I.,  II.,  III.,  denote  that  the  piece  has  been  taken  from  a 
lintel,  and  also  from  what  part,  as  will  easily  be  seen  by  the 
sketch  on  page  379. 

Thus  P.  B.  3  would  signify  that  the  specimen  was  of  quality 
P9  had  been  planed,  and  was  the  third  test  of  this  class. 

On  the  other  hand,  P.  B.  3  II.,  would  signify  in  addition 
that  it  had  been  taken  from  a  lintel,  and  was  a  piece  of  one  of 
the  strips  marked  II.  in  the  sketch. 

The  following  is  a  summary  of  the  breaking-weights  per 
square  inch  of  the  specimens  not  cut  from  the  lintels : 

P.  A.  i 23757  S.  A.  i 24204 

P.  A.  2 21423  S.  A.  2 25258 

P.  A.  3 18938  S.  A.  3 24706 

P.  A.  4...., 21409  

3)74168 


24723 


21382 

P.  B.  i 21756 

P.  B.3 25207 

2)46963 


S.  B.  i 
S.  B,  2 


29574 
23201 


2)52775 


23482  26388 

The  following  are  the  breaking-weights  per  square  inch  of 
the  specimens  cut  from  the  iinteis : — 


P  B. 


'  5 

I 

19651 

6 

I 

20715 

9 

I 

21076 

10 

I 

21483 

4 

II 

19016 

7 

II 

19376 

ii 

II 

22146 

12 

II 

20552 

2 

III 

10594 

(Broke  at 
a  flaw.) 

13 

III 

16141 

I  8 

IV 

10616 

S.  B. 


6 

I 

29124 

7 

I 

28372 

8 

I 

25425 

3 

II 

24704 

4 

II 

29414 

5 

II 

23610 

9 

III 

27523 

10 

III 

18301 

4 

IV 

19616 

TKAffSVERSE  STRENGTH  OF  CAST-IRON. 


381 


All  the  window  lintels  tested  were  of  the  form  shown  in 
the  figure,  and  all  were  supported  at  the  ends  and  loaded  at 
the  middle,  the  span  in  every  case  being  52".  From  the  cut 
it  will  be  seen  that  the  web  varied  in  height,  being  4  inches 
high  above  the  flange  in  the  centre,  and  decreasing  to  2.5  inches 
at  the  ends  over  the  supports. 

The  following  are  the  results  of  the  separate  tests,  where 
tensile  modulus  of  rupture  means  the  outside  fibre  stress  per 
square  inch  on  the  tension  side,  and  compressive  modulus  of 
rupture  that  on  the  compression  side,  both  being  calculated 
from  the  actual  breaking  load  by  the  formula 


f- 
J  ~    '      ' 


Mark  on  Lintel. 

Breaking-Load, 
Lbs. 

Tensile  Modulus  of 
Rupture, 
Ibs.  per  Sq.  In. 

Compressive  Modulus 
of  Rupture, 
Ibs.  per  Sq.  In. 

P.    I 

27220 

26648 

81578 

P.    2 

30520 

29879 

91467 

P.  3 

27200 

26659 

81608 

S.    i 

26750 

26198 

80164 

S.     2 

19850 

19433 

59490 

S.   3 

28670 

28068 

85924 

S.   4 

25120 

24592 

75285 

The  second  series  of  experiments  was  made  by  two  other 
students,  and  an  account  of  the  work  is  given  in  the  same 
article  as  the  former  one. 

The  object  was  to  determine  the  constants  suitable  to  use 
in  the  formulae  for  determining  the  strength  of  the  arms  of 
cast-iron  pulleys  ;  and  also,  incidentally,  to  determine  the  hold- 
ing power  of  keys  and  set-screws. 

Some  old  pulleys  with  curved  arms,  which  had  been  in  use 
at  the  shops,  were  employed  for  these  tests.  They  were  all 


382  APPLIED   MECHANICS. 

about  fifteen  inches  in  diameter,  and  were  bored  for  a  shaft 
IT^  inches  in  diameter. 

Inasmuch  as  this  size  of  shaft  would  not  bear  the  strain 
necessary  to  break  the  arms,  the  hubs  were  bored  out  to  a 
diameter  of  i-j-J-  inches  diameter,  and  key-seated  for  a  key  one- 
half  an  inch  square. 

In  order  to  strengthen  the  hubs  sufficiently,  two  wrought- 
iron  rings  were  shrunk  on  them,  so  as  to  make  it  a  test  of  the 
arms  and  not  of  the  hub. 

The  pulley  under  test  is  keyed  to  a  shaft  which,  in  its  turn, 
is  keyed  to  a  pair  of  castings  supported  by  two  wrought-iron  I- 
beams,  resting  upon  a  pair  of  jack-screws,  by  means  of  which 
the  load  is  applied.  A  wire  rope  is  wound  around  the  rim  of 
the  pulley,  and  leaves  it  in  a  tangential  direction  vertically. 
This  rope  is  connected  with  the  weighing  lever  of  the  machine, 
and  weighs  the  load  applied. 

In  a  number  of  the  experiments  one  arm  gave  way  first, 
and  then  the  unsupported  part  of  the  rim  broke. 

The  breaking-load  of  the  separate  pulleys  was,  of  course, 
determined,  and  then  it  was  sought  to  compute  from  this  the 
value  of  f  from  the  formula 


which  is  the  one  most  commonly  given  for  the  strength  of 
pulley  arms,  and  which  is  based  upon  several  erroneous  assump- 
tions, one  of  which  is  that  the  bending-moment  is  equally 
divided  among  the  several  arms.  In  this  formula 

/=  moment  of  inertia  of  section, 

n  =  number  of  arms, 

y  =  half  depth  of  each  arm  =  distance  from  neutral  axis  to 
outside  fibre, 

x  =  length  of  each  arm  in  a  radial  direction, 

P  =  breaking-load  determined  by  experiment. 


TRANSVERSE    STRENGTH  OF   CAST-IRON. 


383 


The  results  are  given  in  the  following  table,  the  units  being 
inches  and  pounds : 


the 
the 


o   o 

H 

rt    cl 

II 


tly  inc 
broke. 


H 

!i 
ii 


sad  subsequently 
when  the  rim  brok 


.sg  ~| 

S  "5     - 


II  S*53 

£j-  -S:rc3 
13  33  la 


5*1 


8JS 


u       i» 


!!  iflSrf"2 

°§     S^13 

s2   is^-sl. 

O     H 


C-«      J3- 

O       H 


rt 
| 

1    I 


=•3    s- 


«U^3 

O 


I 


.2  "u 


§  § 

Q  S 


XXX 


•P 
x 


L-^C 

X 


X 

<£ 


t-pj    t-bj 
Mps    otjeo 

X      X 


X 

-p 


* 

X 


N|m 
X      X 


•stnay  jo  aaqtnnj^  | 


•souy 


a 


" 


•qnH 


co       co       co 


jo  ssaujpiqi, 


-C    -C 


-^MH       WH 
•  co  co         CO 


•Xannj  jo  -oreia 


IT        rt        2"       "?        iT    ? 


APPLIED  MECHANICS. 


In  the  cases  of  numbers  5,  7,  8,  9,  and  10  some  of  the  arms 
were  not  broken,  the  rims  were  now  broken  off,  and  the  re- 
maining arms  were  tested  separately,  the  pull  being  exerted  by 
a  yoke  hung  over  the  end  of  the  arm,  the  lower  end  being  at- 
tached to  the  link  of  the  machine. 

The  arms  were  always  placed  so  that  the  direction  of  the 
pull  was  tangent  to  the  curve  of  the  rim  at  the  end  of  the  arm. 
The  actual  modulus  of  rupture  was  then  determined  by  calcula- 
tion from  the  experimental  results,  and  is  recorded  in  the 
following  table,  the  units  being  inches  and  pounds  : — 


Number  of 
Arms. 

Dimensions  of  Sec- 
tion at  Fracture  : 
all  elliptical. 

Bend  of  Arm  with 
or  against  Load. 

Modulus  of 
Rupture. 

AverageModulus  of 
Rupture  for  each 
Pulley. 

5  —  i 

i*  X  iJ 

against 

45396 

45396 

7  —  i 

i*     X* 

against 

36802 

7  —  2 

'if  Xf 

against 

39537 

7  —  3 

itfxi 

with 

46407 

40915 

8  —  i 

itt  x  H 

against 

35503 

8  —  2 

iH  x|| 

against 

36091 

8-3 

i«X« 

with 

39939 

8-4 

iHxtt 

with 

42469 

38500 

9  —  1 

i*Xf 

against 

41899 

9  —  2 

'*XH 

against 

44148 

9  —  3 

i*xf 

with 

55442 

47163 

10  I 

if  XH 

against 

54743 

10  2 

I«XH 

against 

5°943 

io  —  3 

i«x« 

against 

38605 

10  —  4 

if  x  ft 

with 

55229 

49880 

—  -^^—  —  —  —  « 

STANDARD    SPECIFICATIONS  FOR   CAST-IRON.          385 

STANDARD     SPECIFICATIONS    FOR    CAST-IRON,    OF     THE    AMERICAN 
SOCIETY   FOR  TESTING   MATERIALS. 

The  standard  specifications  for  cast-iron,  of  the  American 
Society  for  Testing  Materials,  contain  specifications  for  i° 
Foundry  Pig-iron,  2°  Gray  Iron  Castings,  3°  Malleable  Iron 
Castings,  4°  Locomotive  Cylinders,  5°  Cast-iron  Pipe  and  Special 
Castings,  6°  Cast-iron  Car-wheels.  Of  these,  i°,  2°,  and  4°  will 
be  quoted  in  full,  and  extracts  will  be  given  from  5°.  For  the 
remainder  see  the  proceedings  of  the  Society. 

AMERICAN  SOCIETY  FOR  TESTING  MATERIALS. 
SPECIFICATIONS  FOR  FOUNDRY  PIG-IRON. 

ANALYSIS. 
It  is  recommended  that  all  purchases  be  made  by  analysis. 

SAMPLING. 

In  all  contracts  where  pig-iron  is  sold  by  chemical  analysis,  each 
car  load,  or  its  equivalent,  shall  be  considered  as  a  unit.  At  least  one 
pig  shall  be  selected  at  random  from  each  four  tons  of  every  car  load, 
and  so  as  to  fairly  represent  it. 

Drillings  shall  be  taken  so  as  to  fairly  represent  the  fracture-surface 
of  each  pig,  and  the  sample  analysed  shall  consist  of  an  equal  quantity 
of  drillings  from  each  pig,  well  mixed  .and  ground  before  analysis. 

In  case  of  disagreement  between  buyer  and  seller,  an  independent 
analyst,  to  be  mutually  agreed  upon,  shall  be  engaged  to  sample  and 
analyze  the  iron.  In  this  event  one  pig  shall  be  taken  to  represent 
every  two  tons. 

The  cost  of  this  sampling  and  analysis  shall  be  borne  by  the  buyer 
if  the  shipment  is  proved  up  to  specifications,  and  by  the  seller  if  other- 
wise. 

ALLOWANCES  AND  PENALTIES. 

In  all  contracts,  in  the  absence  of  a  definite  understanding  to  the 
contrary,  a  variation  of  10  per  cent  in  silicon,  either  way,  and  of  o.oi 
sulphur,  above  the  standard,  is  allowed. 

A  deficiency  of  over  10  per  cent  and  up  to  20  per  cent,  in  the  silicon, 
subjects  the  shipment  to  a  penalty  of  4  per  cent  of  the  contract  price. 


386 


APPLIED    MECHANICS. 


BASE  ANALYSIS  OF  GRADES. 

In  the  absence  of  specifications,  the  following  numbers,  known  to 
the  trade,  shall  represent  the  appended  analyses  for  standard  grades 
of  foundry  pig-irons,  irrespective  of  fracture,  and  subject  to  allowances 
and  penalty  as  above: 


Grade. 

Per  Cent 

Silicon. 

Per  Cent 
Sulphur 
(Volumetric). 

Per  Cent 

Sulphur 
(Gravimetric). 

No    i   .      .      . 

2-75 

0-035 

0.045 

N  >.    2     .         .         . 

2.25 

0.045 

0-055 

No    3    ... 

i-75 

0-°55 

0.065 

No   4    ... 

1.25 

0.065 

0.075 

PROPOSED  SPECIFICATIONS  FOR  GRAY  IRON  CASTINGS. 
PROCESS  OF  MANUFACTURE. 

Unless  furnace  iron  is  specified,  all  gray  castings  are  understood  to 
be  made  by  the  cupola  process. 

CHEMICAL  PROPERTIES. 
The  sulphur  contents  to  be  as  follows: 

Light  castings not  over  o .  08  per  cent. 

Medium  castings        .      .      .      .     "      "     o.io    "      " 
Heavy  castings "      "     0.12    "      " 

DEFINITION. 

In  dividing  castings  into  light,  medium,  and  heavy  classes,  the 
following  standards  have  been  adopted : 

Castings  having  any  section  less  than  \  of  an  inch  thick  shall  be 
known  as  light  castings. 

Castings  in  which  no  section  is  less  than  2  ins.  thick  shall  be  known 
as  heavy  castings. 

Medium  castings  are  those  not  included  in  the  above  definitions. 
PHYSICAL  PROPERTIES. 

Transverse  Test.     The  minimum  breaking-strength  of  the  "Arbi- 
tration Bar  "  under  transverse  load  shall  not  be  under: 

Light  castings 2500  Ibs. 

Medium  castings 2900    " 

Heavy  castings 3300    " 


STANDARD   SPECIFICATIONS   t-'OR    CAST-IRON. 


387 


In  no  case  shall  the  deflection  be  under  .10  of  an  inch. 

Tensile  Test.     Where  specified,  this  shall  not  run  less  than: 

Light  castings 18000  Ibs.  per  square  inch. 

Medium  castings    ....    21000    "      "        "         " 
Heavy  castings 24000    "      ' '        ' '         " 

THE  " ARBITRATION  BAR"  AND  METHODS  OF  TESTING. 

The  quality  of  the  iron  going  into  castings  under  specification 
shall  be  determined  by  means  of  the  " Arbitration  Bar."  This  is 
a  bar  ij  ins.  in  diameter  and  15  ins.  long.  It  shall  be  prepared  as 
stated  further  on  and  tested  transversely.  The  tensile  test  is  not 
recommended,  but  in  case  it  is  called  for,  the  bar  as  shown  in  Fig.  i, 
and  turned  up  from  any  of  the  broken  pieces  of  the  transverse  test, 
shall  be  used.  The  expense  of  the  tensile  test  shall  fall  on  the  purchaser. 

Two  sets  of  two  bars  shall  be  cast  from  each  heat,  one  set  from  the 
first  and  the  other  set  from  the  last  iron  going  into  the  castings.  Where 


i 


:r 


the  heat  exceeds  twenty  tons,  an  additional  set  of  two  bars  shall  be 
cast  for  each  twenty  tons  or  fraction  thereof  above  this  amount.  In 
case  of  a  change  of  mixture  during  the  heat,  one  set  of  two  bars  shall 
also  be  cast  for  every  mixture  other  than  the  regular  one.  Each  set 
of  two  bars  is  to  go  into  a  single  mold.  The  bars  shall  not  be  rumbled 
or  otherwise  treated,  being  simply  brushed  off  before  testing. 


$88  APPLIED    MECHANICS. 

The  transverse  test  shall  be  made  on  all  the  bars  cast,  with  supports 
12  ins.  apart,  load  applied  at  the  middle,  and  the  deflection  at  rupture 
noted.  One  bar  of  every  two  of  each  set  made  must  fulfill  the  re- 
quirements to  permit  acceptance  of  the  castings  represented. 

The  mold  for  the  bars  is  shown  in  Fig.  2  (not  shown  here).  The 
bottom  of  the  bar  is  iV  of  an  inch  smaller  in  diameter  than  the  top, 
to  allow  for  draft  and  for  the  strain  of  pouring.  The  pattern  shall  not 
be  rapped  before  withdrawing.  The  flask  is  to  be  rammed  up  with 
green  molding-sand,  a  little  damper  than  usual,  well  mixed  and  put 
through  a  No.  8  sieve,  with  a  mixture  of  one  to  twelve  bituminous 
facing.  The  mold  shall  be  rammed  evenly  and  fairly  hard,  thoroughly 
dried  and  not  cast  until  it  is  cold.  The  test-bar  shall  not  be  removed 
from  the  mold  until  cold  enough  to  be  handled. 

SPEED  OF  TESTING. 

The  rate  of  application  of  the  load  shall  be  thirty  seconds  for  a 
deflection  of  .10  of  an  inch. 

* 
SAMPLES  FOR  CHEMICAL  ANALYSIS. 

Borings  from  the  broken  pieces  of  the  "  Arbitration  Bar  "  shall 
be  used  for  the  sulphur  determinations.  One  determination  for  each 
mold  made  shall  be  required.  In  case  of  dispute,  the  standards  of 
the  American  Foundrymen's  Association  shall  be  used  for  comparison. 

FINISH. 

Castings  shall  be  true  to  pattern,  free  from  cracks,  flaws,  and  ex- 
cessive shrinkage.  In  other  respects  they  shall  conform  to  whatever 
points  may  be  specially  agreed  upon. 

INSPECTION. 

The  inspector  shall  have  reasonable  facilities  afforded  him  by  the 
manufacturer  to  satisfy  him  that  the  finished  material  is  furnished  in 
accordance  with  these  specifications.  All  tests  and  inspections  shall, 
as  far  as  possible,  be  made  at  the  place  of  manufacture  prior  to  ship- 
ment. 


STANDARD    SPECIFICATIONS  FOR    CAST-IRON.        389 

SPECIFICATIONS  FOR  LOCOMOTIVE  CYLINDERS. 

PROCESS  OF  MANUFACTURE. 

Locomotive  cylinders  shall  be  made  from  good  quality  of  close- 
grained  gray  iron  cast  in  a  dry  sand  mold. 

CHEMICAL  PROPERTIES. 

Drillings  taken  from  test-pieces  cast  as  hereafter  mentioned  shall 
conform  to  the  following  limits  in  chemical  composition : 

Silicon from  1.25  to  i .  75  per  cent 

Phosphorus not  over     .9      ' '      " 

Sulphur "      "       .10    "      " 

PHYSICAL  PROPERTIES. 

The  minimum  physical  qualities  for  cylinder  iron  shall  be  as 
follows : 

The  ''Arbitration  Test-Bar,"  ij  ins.  in  diameter,  with  supports 
12  ins.  apart  shall  have  a  transverse  strength  not  less  than  30x50  Ibs., 
centrally  appliedj  and  a  deflection  not  less  than  o.io  of  an  inch. 

TEST-PIECES  AND  METHOD  OF  TESTING. 

The  standard  test  shall  be  ij  ins.  in  diameter,  about  14  ins.  long, 
cast  on  end  in  dry  sand.  The  drillings  for  analysis  shall  be  taken 
from  this  test-piece,  but  in  case  of  rejection  of  the  manufacturer  shall 
have  option  of  analyzing  drillings  from  the  bore  of  the  cylinder,  upon 
which  analysis  the  acceptance  or  rejection  of  the  cylinder  shall  be 
based. 

One  test-piece  for  each  cylinder  shall  be  required. 

CHARACTER  OF  CASTINGS. 

Castings  shall  be  smooth,  well  cleaned,  free  from  blow-holes,  shrink- 
age cracks,  or  other  defects,  and  must  finish  to  blue-print  size. 

Each  cylinder  shall  have  cast  on  each  side  of  saddle  manufacturer's 
mark,  serial  number,  date  made,  and  mark  showing  order  number. 

INSPECTOR. 

The  inspector  representing  the  purchaser  shall  have  all  reasonable 
facilities  afforded  to  him  by  the  manufacturer  to  satisfy  himself  that  the 
finished  material  is  furnished  in  accordance  with  these  specifications. 
All  tests  and  inspections  shall  be  made  at  the  place  of  the  manufacturer. 


39°  APPLIED    MECHANICS. 

CAST-IRON    PIPE   AND    SPECIAL   CASTINGS. 

This  specification  is  divided  into  the  following  sections,  viz.:  i° 
Description  of  Pipes,  2°  Allowable  Variation  in  Diameter  of  Pipes  and 
Sockets,  3'°  Allowable  Variation  in  Thickness,  4°  Defective  Spigots  may 
be  Cut,  5°  Special  Castings,  6°  Marking,  7°  Allowable  Percentage  of 
Variation  in  Weight,  8°  Quality  of  Iron,  9°  Tests  of  Material,  10°  Cast- 
ing of  Pipes,  11°  Quality  of  Castings,  12-°  Cleaning  and  Inspection,  13° 
Coating,  14°  Hydrostatic  Test,  15°  Weighing,  16°  Contractor  to  Furnish 
Men  and  Materials,  17°  Power  of  Engineer  to  Inspect,  18°  Inspector 
to  Report,  19°  Castings  to  be  Delivered  Sound  and  Perfect,  20°  Defi- 
nition of  the  Word  Engineer. 

Of  these,  only  sections  8°  and  9°  will  be  quoted  here,  as  follows: 

QUALITY  OF  IRON. 

SECTION  8.  All  pipes  and  special  castings  shall  be  made  of  cast- 
iron  of  good  quality,  and  of  such  character  as  shall  make  the  metal 
of  the  castings  strong,  tough,  and  of  even  grain,  and  soft  enough  to 
satisfactorily  admit  of  drilling  and  cutting.  The  metal  shall  be  made 
without  any  admixture  of  cinder-iron  or  other  inferior  metal,  and  shall 
be  remelted  in  a  cupola  or  air  furnace. 

TESTS  OF  MATERIAL. 

SECTION  9.  Specimen  bars  of  the  metal  used,  each  being  26  inches 
long  by  2  inches  wide  and  i  inch  thick,  shall  be  made  without  charge 
as  often  as  the  engineer  may  direct,  and,  in  default  of  definite  instruc- 
tions, the  contractor  shall  make  and  test  at  least  one  bar  from  each  heat 
or  run  of  metal.  The  bars,  when  placed  flatwise  upon  supports  24 
inches  apart  and  loaded  in  the  centre,  shall  for  pipes  12  inches  or  less 
in  diameter  support  a  load  of  1900  pounds  and  show  a  deflection  of 
not  less  than  .30  of  an  inch  before  breaking,  and  for  pipes  of  sizes  larger 
than  12  inches  shall  support  a  load  of  2000  pounds  and  show  a  deflection 
of  not  less  than  .32  of  an  inch.  The  contractor  shall  have  the  right  to 
make  and  break  three  bars  from  each  heat  or  run  of  metal,  and  the  test 
shall  be  based  upon  the  average  results  of  the  three  bars.  Should 
the  dimensions  of  the  bars  differ  from  those  above  given,  a  proper 
allowance  therefor  shall  be  made  in  the  results  of  the  tests. 


WROUGHT-IRON,  39! 


§  221.  Wrou  gilt-Iron. — Wrought -iron  is  obtained  by  melt- 
ing pig-iron  in  contact  with  iron  ore,  oxidizing,  and  burning  out, 
as  far  as  may  be,  the  carbon,  the  phosphorus,  and  the  silicon. 
In  many  cases,  however,  the  charge  consists  largely  of  wrought- 
iron  or. steel  scrap,  and  cast-iron  borings. 

The  process  is  commonly  carried  on  in  a  puddling  furnace, 
where  an  oxidizing  flame  is  passed  over  the  melted  pig-iron. 

As  the  heat  is  not  sufficiently  intense  to  melt  the  wrought- 
iron  produced,  the  metal  is  left  in  a  plastic  condition,  full  of 
bubbles  and  holes,  which  contain  considerable  slag.  It  is  then 
squeezed,  and  rolled  or  hammered,  to  eliminate,  as  far  as  possible, 
the  slag,  and  to  weld  the  iron  into  a  solid  mass. 

The  result  of  this  first  rolling  is  known  as  muck-bar,  and  must 
be  "piled,"  heated,  and  rolled  or  hammered  at  least  once  more 
before  it  is  suitable  for  use  in  construction. 

In  making  the  piles,  while  muck-bar  is  sometimes  used 
exclusively,  a  considerable  part,  and  often  the  greater  part,  is 
made  of  scrap. 

Wrought-iron  is  thus,  throughout  its  manufacture,  a  series 
of  welds.  Moreover,  wherever  slag  is  present,  these  welds  cannot 
be  perfect.  It  is  also  subject  to  the  impurities  of  the  cast-iron 
from  which  it  is  made.  Thus,  the  presence  of  sulphur  makes 
it  red-short,  or  brittle  when  hot;  and  the  presence  of  phosphorus 
makes  it  cold-short,  or  brittle  when  cold. 

It  cannot,  like  cast-iron,  be  melted  and  run  into  moulds; 
but  it  can  be  easily  welded  by  the  ordinary  methods 

Wrought-iron  is  much  more  capable  of  bearing  a  tensile  or 
transverse  stress  than  cast-iron:  it  is  tougher,  it  stretches  more, 
and  gives  more  warning  before  fracture.  At  one  time  cast-iron 
was  the  principal  structural  material,  but  it  was  soon  displaced 
by  wrought-iron,  which  became  the  principal  metal  used  in 
construction,  but  now,  since  the  modern  methods  of  steel-making 
supply  a  more  homogeneous  product  at  a  cheaper  price,  wrought- 
'iron  has  been  superseded  by  mild  steel  in  most  pieces  used  in 
construction. 


392  APPLIED   MECHANICS. 

Wrought-iron  is  also  expected  to  withstand  a  great  many 
trials  that  would  seriously  injure  cast-iron:  thus,  two  pieces 
of  wrought-iron  are  generally  united  together  by  riveting;  the 
holes  for  the  rivets  have  to  be  punched  or  drilled,  and  then  the 
rivets  have  to  be  hammered;  the  entire  process  tending  to  injure 
the  iron.  Wrought-iron  has  to  withstand  flanging,  and  is  liable 
to  severe  shocks  when  in  use;  as,  for  instance,  those  that  occur 
from  the  changes  of  temperature  in  the  different  parts  of  a  steam- 
boiler. 

The  following  references  to  a  large  number  of  tests  of  wrought- 
iron  will  be  given  : 

i°.  Eaton  Hodgkinson:    (a)  Report  of  Commissioners  on  the  Applica- 

tion of  Iron  to  Railway  Structures. 
(b)  London  Philosophical  Transactions.     1840. 
2°.  William  H.  Barlow:   Barlow's  Strength  of  Materials. 
3°.  Sir  William  Fairbairn:  On  the  Application  of  Cast  and  Wrought 

Iron  to  Building  Purposes. 
4°.  Franklin  Institute  Committee:    Report  of  the  Committee  of  the 

Franklin    Institute.      In    the    Franklin    Institute    Journal    of 


5°.  L.  A.  Beardslee,  Commander  U.S.N.  :  Experiments  on  the  Strength 
of  Wrought-iron  and  of  Chain  Cables.  Revised  and  enlarged 
by  William  Kent,  M.E.,  or  Executive  Document  98,  45th 
Congress,  as  stated  below. 

6°.  David  Kirkaldy:  Experiments  on  Wrought-iron  and  Steel. 

7°.  G.  Bouscaren:  Report  on  the  Progress  of  Work  on  the  Cincinnati 
Southern  Railway,  by  Thomas  D.  Lovett.  Nov.  i,  1875. 

8°.  Tests  of  Metals  made  at  Watertown  Arsenal.  Of  these  the  first 
two  volumes  were  published  before  1881,  and  since  that 
time  one  volume  has  been  published  every  year.  Nearly  all 
of  them  contain  tests  of  wrought-iron  and  a  great  many  of 
them  contain  tests  of  full-size  pieces  of  wrought-iron. 

9°.  A.  Wohler:  (a)  Die  Festigkeits  versuche  mit  Eisen  und  Stahl. 

(b)   Strength  and  Determination  of  the  Dimensions  of  Structures 


TENSILE   STRENGTH  OF    WROUGHT-IRON.  393 

of  Iron  and  Steel,  by  Dr.  Phil.  Jacob  J.  Weyrauch.     Translated 

by  Professor  Dubois. 
10°.  Technology  Quarterly,  Vol.  VII.  No.  2,  Vol.  VIII.  No.  3,  Vol. 

IX.  Nos.  2  and  3,  and  Vol.  X.  No.  4. 
11°.  Mitt,  der  Materialpriifungsaustalt  in  Zurich. 
12°.  Mitt,  aus  dem  Mech.  Tech.  Lab.  in  Berlin. 
13°.  Mitt,  aus  dem  Mech.  Tech.  Lab.  in  Miinchen. 

§  222.  Tensile  Strength  of  Wrought-iron. — About  the 
year  1840  was  published  the  report  of  the  Commission  appointed 
by  the  British  Government  to  investigate  the  application  of  iron 
to  railway  structures.  While  a  number  of  tests  of  iron  had  been 
previously  made,  this  work  may  properly  be  regarded  as  having 
been  the  first  investigation  of  the  kind  that  was  at  all  thorough. 
At  that  time  cast-iron  was  the  metal  most  used  in  construction, 
and  hence  the  greater  part  of  the  work  of  the  Commission  was 
devoted  to  a  study  of  that  metal.  They  made,  however,  a  number 
of  tests  of  wrought-iron,  which,  though  they  were  of  the  greatest 
value  at  the  time,  and  still  have  some  value,  will  not  be  quoted 
here. 

At  about  that  time  the  use  of  wrought-iron  began  to  increase 
at  a  rapid  rate,  the  necessary  appliances  were  introduced  to  roll 
it  into  I  beams,  channel-irons,  angle-irons,  and  other  shapes, 
and  it  began  to  displace  cast-iron  for  one  after  another  purpose 
until  it  came  to  be  the  metal  most  extensively  used  in  construction, 
both  in  the  case  of  structures  and  machines. 

At  first  the  chief  desideratum  was  assumed  to  be  that  it 
should  have  a  high  tensile  strength,  and  scarcely  any  attention 
was  paid  to  its  ductility. 

About  1865,  however,  engineers  began  to  realize  that  duc- 
tility is  an  all-important  property  of  a  metal  to  be  used  in 
construction,  and  that  this  is  not  necessarily  and  not  generally 
obtainable  with  a  very  high  tensile  strength.  The  most 


394  APPLIED    MECHANICS. 

prominent  advocate,  at  that  time,  of  the  importance  of  duc- 
tility was  David  Kirkaldy,  who  published  a  book,  entitled 
"  Experiments  on  Wrought  Iron  and  Steel,"  containing  the 
results  of  his  tests  down  to  1866. 

In  the  early  part  of  his  book  will  be  found  a  summary  of 
what  had  been  done  by  earlier  experimenters  in  this  line. 

Kirkaldy  tested  a  large  number  of  English  irons,  determin- 
ing both  their  breaking-strengths  and  their  ductility. 

In  the  light  of  the  results  obtained  by  him,  he  proceeded 
to  draw  up  his  famous  sixty-six  conclusions. 

These  sixty-six  conclusions  will  not  be  quoted  here,  but 
the  following  statement  will  be  made  regarding  the  main 
results  of  his  work  : 

i°.  He  proved  that  the  results  obtained  by  testing  grooved 
specimens  (or  specimens  of  such  form  as  to  interfere  with  the 
flow  of  the  metal  while  under  test)  did  not  indicate  correctly 
the  quality  of  the  metal,  but  that  such  specimens  should  be 
used  as  did  not  interfere  with  the  flow  of  the  metal.. 

2°.  He  advocated,  with  all  the  earnestness  of  which  he  was 
capable,  the  conclusion  that  it  was  of  the  greatest  importance 
that  all  wrought-iron  used  in  construction  should  have  a  good 
ductility,  and,  in  his  tests,  he  adopted  five  different  methods 
of  measuring  ductility. 

These  methods  are  :  i°.  Contraction  of  area  at  fracture  per 
cent ;  2°.  Ultimate  elongation  per  cent ;  3°.  Breaking-strength 
per  square  inch  of  fractured  area ;  4°.  Contraction  of  stretched 
area  per  cent,  i.e.,  the  contraction  of  area  attained  when  the 
maximum  load  is  first  reached;  5°.  Breaking-weight  per  square 
inch  of  stretched  area.  Of  these  only  two  are  used  at  the  present 
time,  the  first  and  second,  and  they  serve  as  measures  of 
ductility.  These  two  are  the  principal  conclusions  from  Kir- 
kaldy's  tests,  though  he  cites  a  great  many  more,  one  of  the 
principal  of  them  being  his  conclusion  regarding  so-called  cold 
crystallization,  which  will  be  mentioned  later. 


SPECIFICATIONS  FOR    WROUGHT-IRON. 


395 


Tests  of  the  tensile  strength  of  wrought-iron  may  be  divided 
into  two  classes:  i°  those  made  mainly  for  the  purpose  of  deter- 
mining the  quality  of  the  material,  and  2°  those  made  upon  such 
full-size  pieces  as  are  used  in  practice  to  resist  tension. 

The  tests  of  the  first  class  are  made  upon  small  specimens, 
and,  in  order  that  the  results  may  be  comparable,  the  use  of 
standard  forms  and  dimensions  is,  generally,  a  desideratum. 
The  specifications  for  wrought-iron  of  the  American  Society  for 
Testing  Materials  will  be  given  first,  as  they  refer  to  the  kind 
of  wrought-iron  that  is  in  most  common  use,  and  then  some 
other  tensile  tests  of  various  kinds  of  wrought-iron  in  small  pieces 
will  be  given.  Subsequently  tests  of  wrought-iron  eye-bars  will 
be  quoted. 

AMERICAN  SOCIETY  FOR  TESTING  MATERIALS. 
SPECIFICATIONS  FOR  WROUGHT-IRON. 

PROCESS  OF  MANUFACTURE. 

1.  Wrought-iron  shall  be  made  by  the  puddling  process  or  rolled 
from  fagots  or  piles   made   from  wrought-iron  scrap,   alone  or  with 
muck-bar  added. 

PHYSICAL  PROPERTIES. 

2.  The  minimum  physical  qualities  required  in  the  four  classes  of 
wrought-iron  shall  be  as  follows : 


Stay-bolt 
Iron. 

Merchant 
Iron. 
Grade  "A." 

Merchant 
Iron. 
Grade  "B." 

Merchant 
Iron, 
Grade  "C." 

Tensile   strength,  pounds 

per  square  inch  . 

46000 

50000 

48000 

48000 

Yield-point,    pounds    per 

square  inch 

25000 

25000 

25000 

25000 

Elongation,  per  cent  in  8 

inches     

28 

25 

20 

2O 

3.  In  sections  weighing  less  than  0.654  pound  per  lineal  foot,  the 
percentage  of  elongation  required  in  the  four  classes  specified  in  para- 


39^  APPLIED    MECHANICS. 

graph  No.  2  shall  be  12  per  cent.,  15  per  cent.,  18  per  cent.,  and 
21  per  cent.,  respectively. 

4.  The  four  classes  of  iron  when  nicked  and  tested  as  described  in 
paragraph  No.  9  shall  show  the  following  fracture : 

(a)  Stay-bolt  iron,  a  long,  clean,  silky  fibre,  free  from  slag  or  dirt 
and  wholly  fibrous,  being  practically  free  from  crystalline  spots. 

(b)  Merchant  iron,  Grade  "A,"  a  long,  clean,  silky  fibre,  free  from 
slag  or  dirt  or  any  course  crystalline  spots.     A  few  fine  crystalline 
spots  may  be  tolerated,  provided  they  do  not  in  the  aggregate  exceed 
10  per  cent  of  the  sectional  area  of  the  bar. 

(c)  Merchant  iron,  Grade  "B,"  a  generally  fibrous  fracture,  free 
from  coarse  crystalline  spots.     Not  over  10  per  cent   of  the  fractured 
surface  shall  be  granular. 

(d)  Merchant  iron,  Grade  "C,"  a  generally  fibrous  fracture,  free 
from  coarse  crystalline  spots.     Not  over  15  per  cent  of   the  fractured 
surface  shall  be  granular. 

5.  The  four  classes  of  iron,  when  tested  as  described  in  paragraph 
No.  10,  shall  conform  to  the  following  bending  tests: 

(e)  Stay-bolt  iron,  a  piece  of  stay-bolt  iron  about  24  inches  long, 
shall  bend  in  the  middle  through  180°  flat  on  itself,  and  then  bend  in 
the  middle  through   180°  flat  on  itself  in  a  plane  at  a  right  angle  to 
the  former    direction    without    a    fracture   on   outside   of   the    bent 
portions.     Another  specimen  with  a  thread  cut  over  the  entire  length 
shall  stand  this  double  bending  without  showing  deep  cracks  in  the 
threads. 

(/)  Merchant  iron,  Grade  "A,"  shall  bend  cold  180°  flat  on  itself, 
without  fracture  on  outside  of  the  bent  portion. 

(g)  Merchant  iron,  Grade  "B,"  shall  bend  cold  180°  around  a 
diameter  equal  to  the  thickness  of  the  tested  specimen,  without  fracture 
on  outside  of  bent  portion. 

(h)  Merchant  iron,  Grade  "C,"  shall  bend  cold  180°  around  a 
diameter  equal  to  twice  the  thickness  of  the  specimen  tested,  without 
fracture  on  outside  of  the  bent  portion. 

6.  The  four  classes  of  iron  when  tested  as  described  in  paragraph 
No.  n,  shall  conform  to  the  following  hot  bending  tests: 

(i)  Stay-olt  iron,  shall  bend  through  180°  flat  on  itself,  without 


SPECIFICATIONS  FOR    WROUGHT-IRON.  397 

showing  cracks  or  flaws.  A  similar  specimen  heated  to  a  yellow  heat 
and  suddenly  quenched  in  water  between  80°  and  90°  F.  shall  bend, 
without  hammering  on  the  bend,  180°  flat  on  itself,  without  showing 
cracks  or  flaws. 

(/)  Merchant  iron,  Grade  "A,"  shall  bend  through  180°  flat  on 
itself,  without  showing  cracks  or  flaws.  A  similar  specimen  heated 
to  a  yellow  heat  and  suddenly  quenched  in  water  between  80°  and 
90°  F.  shall  bend,  without  hammering  on  the  bend,  180°  flat  on  itself, 
without  showing  cracks  or  flaws.  A  similar  specimen  heated  to  a  bright- 
red  heat  shall  be  split  at  the  end  and  each  part  bent  back  through  an 
angle  of  180°.  It  will  also  be  punched  and  expanded  by  drifts  until 
a  round  hole  is  formed  whose  diameter  is  not  less  than  nine-tenths  of 
the  diameter  of  the  rod  or  width  of  the  bar.  Any  extension  of  the 
original  split  or  indications  of  fracture,  cracks,  or  flaws  developed  by 
the  above  tests  will  be  sufficient  cause  for  the  rejection  of  the  lot  rep- 
resented by  that  rod  or  bar. 

(k)  Merchant  iron,  Grade  "B,"  shall  bend  through  180°  flat  on 
itself,  without  showing  cracks  or  flaws. 

(/)  Merchant  iron,  Grade  "C,"  shall  bend  sharply  to  a  right  angle, 
without  showing  cracks  or  flaws. 

7.  Stay-bolt  iron  shall  permit  of  the  cutting  of  a  clean  sharp  thread 
and  be  rolled  true  to  gauges  desired,  so  as  not  to  jam  in  the  threading 
dies. 

TEST  PIECES  AND  METHODS  OF  TESTING. 

8.  Whenever  possible,  iron  shall  be  tested  in  full  size  as  rolled,  to 
determine  the  physical  qualities  specified  in  paragraphs  Nos.  2  and  3, 
the  elongation  being  measured  on  an  eight  inch  (8")  gauged  length. 
In  flats  and  shapes  too  large  to  test  as  rolled,  the  standard  test  specimen 
shall  be  one  and  one-half  inches  (ii")  wide  and  eight  inches  (8") 
gauged  length. 

In  large  rounds,  the  standard  test  specimen  of  two  inches  (2") 
gauged  length  shall  be  used;  the  center  of  this  specimen  shall  be  half- 
way between  the  center  and  outside  of  the  round.  Sketches  of  these 
two  standard  test  specimens  are  as  follows: 


39*  APPLIED   MECHANICS. 


. 4*: --- \ 


I " 


jt -18-about \ 

PIECE  TO  BE  OF  SAME  THICKNESS  AS  T&E  PLATE. 


9.  Nicking  tests  shall  be  made  on  specimens  cut  from  the  iron  as 
rolled.     The  specimen  shall  be  slightly  and  evenly  nicked  on  one  side 
and  bent  back  at  this  point  through  an  angle  of  180°  by  a  succession  of 
light  blows. 

10.  Cold  bending  tests  shall  be  .made  on  specimens  cut  from  the 
bar  as  rolled.     The  specimen  shall  be  bent  through  an  angle  of  180° 
by  pressure  or  by  a  succession  of  light  blows. 

11.  Hot  bending  tests  shall  be  made  on  specimens  cut  from  the 
bar  as  rolled.     The  specimens,  heated  to  a  bright  red  heat,  shall  be 
bent  through  an  angle  of  180°  by  pressure  or  by  a  succession  of  light 
blows  and  without  hammering  directly  on  the  bend. 

If  desired,  a  similar  bar  of  any  of  the  four  classes  of  iron  shall  be 
worked  and  welded  in  the  ordinary  manner  without  showing  signs  of 
red  shortness. 

12.  The  yield -point  specified  in    paragraph  No.  2  shall  be  deter- 
mined by  the  careful  observation  of  the  drop  of  the  beam  or  halt  in 
the  gauge  of  the  testing-machine. 


TESTS   OF  COMMANDER  BEARDSLEE.  399 

FINISH. 

13.  All  wrought-iron  must   be  practically  straight,  smooth,  free 
from  cinder  spots  or  injurious  flaws,  buckles,  blisters  or  cracks. 

In  round  iron,  sizes  must  conform  to  the  Standard  Limit  gauge 
as  adopted  by  the  Master  Car  Builders'  Association  in  November, 

1883. 

INSPECTION. 

14.  Inspectors  representing  the  purchasers  shall  have  all  reason- 
able facilities  afforded  them  by  the  manufacturer  to  satisfy  them  that 
the  finished  material  is  furnished  in  accordance  with  these  specifications. 
All  tests  and  inspections  shall  be  made  at  the  place  of  manufacture 
prior  to  shipment. 


TESTS  OF   COMMANDER  BEARDSLEE. 

One  of  the  most  valuable  sets  of  tests  of  wrought-iron  is  that 
obtained  by  committees  D,  H,  and  M  of  the  Board  appointed 
by  the  United  States  Government  to  test  iron  and  steel;  the 
special  duties  of  these  committees  being  to  test  such  iron  as  would 
be  used  in  chain-cable,  and  the  chain-cable  itself.  The  chairman 
of  these  three  committees,  which  were  consolidated  into  one,  was 
Commander  L  .A.  Beardslee  of  the  United  States  Navy.  The 
full  account  of  the  tests  is  to  be  found  in  Executive  Document 
98,  45th  Congress,  second  session;  and  an  abridged  account  of 
them  was  published  by  William  Kent,  as  has  been  already 
mentioned. 

The  samples  of  bar-iron  tested  were  round,  and  varied  from 
one  inch  to  four  inches  in  diameter. 


AOO  APPLIED    MECHANICS. 

Certain  conclusions  which  they  reached  refer  to  all  kinds- 
of  wrought-iron,  and  will  be  given  here  before  giving  a  table  of 
the  results  of  the  tests. 

i°.  Kirkaldy  considers  the  breaking-strength  per  square 
inch  of  fractured  area  as  the  main  criterion  by  which  to  deter- 
mine the  merits  of  a  piece  of  iron  or  steel.  Commander 
Beardslee,  on  the  other  hand,  thinks  that  a  better  criterion  is 
what  he  calls  the  "tensile  limit;"  i.e.,  the  maximum  load  the 
piece  sustains  divided  by  the  area  of  the  smallest  section  when 
that  load  is  on,  i.e.,  just  before  the  load  ceases  to  increase  in 
the  testing-machine. 

2°.  Kirkaldy  had  already  called  attention  to  the  fact  that 
the  tensile  strength  of  a  specimen  is  very  much  affected  by  its 
shape,  and  that,  in  a  specimen  where  the  shape  is  such  that 
the  length  of  that  part  which  has  the  smallest  cross-section  is 
practically  zero  (as  is  the  case  when  a  groove  is  cut  around 
the  specimen),  the  breaking-strength  is  greater  than  it  is  when 
this  portion  is  long ;  the  excess  being  in  some  cases  as  much 
as  33  per  cent. 

Commander  Beardslee  undertook,  by  actually  testing  speci- 
mens whose  smallest  areas  varied  in  length,  to  determine  what 
must  be  the  least  length  of  that  part  of  the  specimen  whose 
cross-section  area  is  smallest,  in  order  that  the  tensile  strength 
may  not  be  greater  than  with  a  long  specimen.  The  conclusion 
reached  was,  that  no  test-piece  should  be  less  than  one-half  inch 
in  diameter,  and  that  the  length  should  never  be  less  than  four 
diameters  ;  while  a  length  of  five  or  six  diameters  is  necessary 
with  soft  and  ductile  metal  in  order  to  insure  correct  results. 
The  following  results  of  testing  steel  are  given  in  Mr.  Kent's 
book,  as  confirming  the  same  rule  in  the  case  of  steel.  The 
tests  were  made  upon  Bessemer  steel  by  Col.  Wilmot  at  the 
Woolwich  arsenal. 


TESTS   OF  COMMANDER  BEARDSLEE. 


4OI 


Tensile  Strength. 

Pounds  per 
Square  Inch. 

Highest  
Lowest 

162974 

Average  .  .  .  .  . 
Highest  
T  owpst 

153677 
123165 

Average  

103255 
114460 

3°.  Commander  Beardslee  also  noticed  that  rods  of  certain 
diameters  of  the  same  kind  of  iron  bore  less  in  proportion  than 
rods  of  other  diameters ;  and,  after  searching  carefully  for  the 
reason,  he  found  it  to  lie  in  the  proportion  between  the  diam- 
eter of  the  rod  and  the  size  of  the  pile  from  which  it  is 
rolled.  The  following  examples  are  given  :  — 

ij-in.  diameter,  6.62%  of  pile,  56543  Ibs.  per  sq.  in.  tensile  strength. 


If 
I* 
If 
'I 


8.i8% 

tt 

56478     ' 

9.90% 

" 

54277     " 

11.78% 

tt 

5355°   " 

7.68% 

" 

56344   " 

8.90% 

tt 

55018   " 

10.22% 

(i 

54034   " 

II-63% 

(i 

51848   " 

He  therefore  claims,  that,  in  any  set  of  tests  of  round  iron, 
it  is  necessary  to  give  the  diameter  of  the  rod  tested,  and  not 
merely  the  breaking-strength  per  square  inch. 

4°.  He  gives  evidence  to  show,  that  if  a  bar  is  under-heated, 
it  will  have  an  unduly  high  tenacity  and  elastic  limit ;  and  that 
if  it  is  over-heated,  the  reverse  will  be  the  case. 


402 


APPLIED   MECHANICS 


5°.  The  discovery  was  made  independently  by  Commander 
Beardslee  and  Professor  Thurston,  that  wrought-iron,  after 
having  been  subjected  to  its  ultimate  tensile  strength  without 
breaking  it,  would,  if  relieved  of  its  load  and  allowed  to  rest, 
have  its  breaking-strength  and  its  limit  of  elasticity  increased. 

His  experiments  show  that  the  increase  is  in  irons  of  a 
fibrous  and  ductile  nature,  rather  than  in  brittle  and  steely 
ones ;  hence  the  latter  class  would  be  but  little  benefited  by 
the  action  of  this  law. 

The  most  characteristic  table  regarding  this  matter  is  the 
following  :— 

EFFECT   OF   EIGHTEEN    HOURS'  REST   ON   IRONS   OF  WIDELY   DIFFER 
ENT   CHARACTERS. 


I 

Ultimate  Strength 

i 
i 

per  Square  Inch. 

i 

"O                1 

First 

Second 

JxGXEl&rJCS* 

Strain. 

Strain. 

Boiler  iron     .     .     . 

48600 

56500 

Not  broken. 

tt        tt 

49800 

57000 

Broken  \ 

tt        (t 

49800 

58000 

Broken  1   Average  gain, 

H                ft 

48100 

54400 

Broken   f                   15.8%. 

(I           11 

48150 

5555° 

Broken  J 

Contract  chain  iron, 

50200 

54000 

Broken         *j 

(t           ii        « 

50250 

53200 

Not  broken   1   Average 

d           ti        tt 

50700 

553oo 

Not  broken  j*          gain, 

n           t(        ft 

49600 

52900 

Not  broken  j              6.4%. 

ft           tt        a 

51200 

52800 

Not  broken  ) 

Iron  K     .     .     .     . 
ft     ft 

58800       64500 

59000  !  65800 

Broken  ^ 
Broken   I  Average  ^ 

ft     tt 

56400  j  60600 

Broken  J                    9-4%- 

j 

CHAIN   CABLE.  403 


§  233.  Chain  Cable.  —  The  most  thorough  set  of  tests  of  the 
strength  of  chain  cable  is  that  made  by  Commander  Beardslee 
for  the  United-States  government,  an  account  of  which  may  be 
found  either  in  the  report  already  referred  to,  or  in  the  abridg- 
ment by  William  Kent. 

In  this  report  are  to  be  found  a  number  of  conclusions, 
some  of  which  are  as  follows  :  — 

i°.  That  cables  made  of  studded  links  (i.e.,  links  with  a 
cast-iron  stud,  to  keep  the  sides  apart)  are  weaker  than  open- 
link  cables. 

2°.  That  the  welding  of  the  links  is  a  source  of  weakness ; 
the  amount  of  loss  of  strength  from  this  cause  being  a  very 
uncertain  quantity,  depending  partly  on  the  suitability  of  the 
iron  for  welding,  and  partly  on  the  skill  of  the  chain-welder. 

3°.  That  an  iron  which  has  a  high  tensile  strength  does  not 
necessarily  make  a  good  iron  for  cables.  Of  the  irons  tested, 
those  that  made  the  strongest  cables  were  irons  with  about 
51000  Ibs.  tensile  strength. 

4°.  The  greatest  strength  possible  to  realize  in  a  cable  per 
square  inch  of  the  bar  from  which  it  is  made  being  200  per 
cent  of  that  of  the  bar-iron  from  which  it  was  made,  the  cables 
tested  varied  from  155  to  185  per  cent  of  that  of  the  bar- 
iron. 

5°.  The  Admiralty  rule  for  proving  chain  cables,  by  which 
they  are  subjected  to  a  load  in  excess  of  their  elastic  limit, 
is  objected  to,  as  liable  to  injure  the  cable :  and  the  report 
suggests,  in  its  place,  a  lower  set  of  proving-strengths,  as  given 
in  the  following  table ;  the  Admiralty  proving-strengths  being 
ilso  given  in  the  table. 

In  these  recommendations,  account  is  taken  of  the  different 
proportion  of  strength  of  different  size  bars  as  they  come  from 
th:  rolls,  also  no  proving-stress  is  recommended  greater  than 
50  per  cent  of  the  strength  of  the  weakest  link,  and  45.5  per 
cent  rf  the  strongest ;  v/hereas  in  the  Admiralty  tests,  66.2 


404 


APPLIED  MECHANICS. 


per  cent  of  the  strength  of  the  weakest,  and  60.3  per  cent  of 
the  strongest,  is  sometimes  used. 

For  the  details  of  this  investigation,  see  the  report,  Execu- 
tive Document  No.  98,  45th  Congress,  second  session,  or  the 
abridgment  already  referred  to. 


Diameter  of 
Iron, 
in  inches. 

Recommended 
Proving-Strains. 

Admiralty 
Proving-Strains. 

Diameter  of 
Iron, 
in  inches. 

Recommended 
Proving-Strains. 

Admiralty 
Proving-Strains. 

2 

121737 

161280 

I* 

66138 

83317 

lit 

114806 

I5I357 

If 

60920 

76230 

I  8 

108058 

I4I750 

i-iV 

55903 

69457 

lit 

101499 

I32457 

li 

5I084 

63000 

If 

95128 

123480 

IA 

46468 

56857 

Itt 

88947 

114817 

It 

42053 

5I03° 

If 

82956 

106470 

nV 

37820 

455*7 

I* 

77159 

98437 

i 

33840 

40320 

it 

7I550 

90720 

While  steel  long  ago  displaced  wrought-iron  for  boiler-plate, 
and  while  steel  I  beams,  channel-bars,  angle-irons,  and  other 
shapes,  as  well  as  eye-bars,  have,  of  late  years,  displaced 
wrought-iron  to  a  very  great  extent,  nevertheless  wrought-iron 
is  still  very  extensively  used,  and  for  a  great  variety  of  struc- 
tural purposes. 

For  wrought-iron  to  be  used  in  construction,  ductility, 
homogeneity,  and  often  weldability  are  the  great  desiderata, 
together  with  as  large  a  tensile  strength  as  is  consistent  with 
these.  As  to  the  requirements  made  by  different  engineers  for 
wrought-iron  for  structural  purposes,  the  minimum  tensile 
strength  called  for  varies  from  about  46000  to  about  50000 
pounds  per  square  inch,  with  ultimate  elongations  varying  from 
15$  to  30$  in  8  inches,  according  to  the  purpose  for  which  it  is 
wanted.  It  is  also  very  common,  when  good  iron  is  wanted, 


CHAIN   CABLE. 


405 


to  insist  that  it  shall  not  be  made  of  scrap.  The  following 
tables  of  tensile  tests  of  wrought  iron  of  various  kinds  will 
show  what  results  can  be  obtained. 


Norway  Iron. 

Burden's  Best. 

•od 

c 

c 

•e"a 

d 

a 

o"". 

«-.'". 

4> 

o"" 

4-r"! 

-8 

u  . 

1& 

o  a 

1  5 

If 

it 

i 

1? 

o  a 

o  u 
c  w 
.2 
5  jf 

«"  >> 
&  G 

11 

Is' 

u 

|| 

"5  «! 

i! 

u 

11 

Is 

5 

s 

£ 

s 

5 

s 

5 

OS 

s 

•75 

48390 

23620 

62.6 

30090000 

.76 

53566 

27554 

57-6 

29175000 

•75 

46340 

21160 

62.7 

30780000 

•75 

50023 

26030 

49-8 

30643000 

•75 

48280 

28030 

62.6 

29020000 

•76 

47724 

25350 

47-6 

30310000 

•77 

45160 

20400 

68.8 

27388000 

•77 

46772 

24700 

45-2 

28347000 

•75 

46063 

19240 

68.6 

27666000 

•77 

46600 

22550 

46.2 

29528000 

•77 

44490 

20510 

67-5 

28452000 

•77 

47395 

22550 

46.2 

28347000 

•74 

43233 

22079 

70.5 

29026000 

•77 

47963 

22695 

48.6 

29475000 

•75 

43470 

19400 

75-5 

26700000 

•77 

47860 

26948 

46.4 

26948000 

•73 

38950 

22030 

72.3 

30140000 

•77 

475CQ 

26927 

42-3 

28435000 

•74 

43240 

21970 

75-2 

27726000 

•76 

47610 

23036 

53-  l 

29551000 

•74 

44564 

21970 

73.8 

28663000 

•77 

49238 

22725 

49-2 

27470000 

•74 

43860 

19658 

75-o 

18000000 

•76 

50037 

27700 

53-6 

29251000 

i  .00 

41620 

15560 

7°-3 

27295000 

•76 

48538 

27224 

48.8 

29355000 

•75 

42215 

68.6 

29292000 

•76 

50060 

23201 

S3-0 

31028000 

•75 

42033 

19239 

62.4 

29729000 

.76 

49143 

23240 

5°-4 

30438000 

.76 

4'574 

14328 

69-5 

27450000 

.76 

48655 

23414 

49.6 

30062000 

.76 
•75 

41574 
426^6 

16531 
19240 

68.7 
59-° 

29098000 
31785000 

•76 
•  76 

47220 
47090 

22880 
23020 

53-4 

54-i 

29969000 
33657000 

•75 

41875 

16978 

70.1 

30487000 

.76 

49690 

27480 

53-3 

29614000 

•75 

43396 

19112 

59-3 

28000000 

.76 

47430 

22950 

51-8 

29443000 

•74 

39210 

15216 

73'2 

30294000 

•76 

4795° 

23000 

57-o 

29504000 

•74 

12603 

70.5 

28810000 

.76 

22892 

45-8 

28779000 

•74 

39896 

15187 

69.7 

31153000 

•77 

49411 

18420 

46.6 

30112000 

•74 

39156 

16123 

76.4 

29807000 

.76 

49660 

23186 

51-3 

30160000 

•74 

41030 

17490 

69.8 

29310000 

•76 

48055 

20940 

56.7 

28809000 

•75 

41180 

18000 

72.5 

31073000 

•77 

49026 

22578 

40.5 

27292000 

•74 

42320 

19660 

68.0 

30834000 

•76 

47220 

23060 

51-2 

33710000 

•74 

43913 

198:53 

69.8 

26970000 

.76 

5OI49 

20940 

41.5 

27450000 

•74 

42102 

191.81 

78.3 

29127000 

•75 

48553 

23767 

74-3 

31124000 

•74 

39698 

17638 

70-5 

30023000 

49350 

21503 

66.5 

31793000 

•73 

43187 

17846 

68.6 

28553000 

.76 

50083 

20940 

29097000 

•73 

40669 

17820 

73-8 

30159000 

•76 

47019 

23140 

51  -4 

29978000 

•73 

39348 

16593 

69-3 

29518000 

•76 

47504 

20942 

53-2 

28527000 

•73 

39671 

12987 

78.1 

28861000 

•76 

47747 

20942 

49-5 

29874000 

•75 

39951 

16886 

77.2 

30020000 

.76 

50927 

23453 

46.9 

28350000 

•74 

41093 

16400 

74-1 

28634000 

•75 

51269 

21182 

51  -9 

32551000 

•74 

40192 

14053 

76.3 

28627000 

•75 

50930 

23770 

53-7 

29293000 

•73 

44470 

16844 

73-4 

31114000 

•76 

50083 

23146 

45  •  - 

29097000 

•74 

41940 

17523 

78.5 

29373000 

•75 

48168 

23767 

55-6 

29879000 

•74 

42531 

16449 

7°.S 

30410000 

•76 

49500 

26500 

55-o 

3  i  600000 

•75 

48400 

27200 

46.2 

28700000 

•75 

47600 

27200 

50.1 

29300000 

•77 

47200 

23600 

56.1 

27800000 

.76 

46700 

24200 

41.8 

29700000 

•77  . 

45600 

23600 

54-5 

28200000 

406 


APPLIED    MECHANICS. 


Refined  Iron. 

Wrought-iron  Wire. 

|a 

a 

c 
u 

"rt'.S 

,H 

C 
V 

% 

J  a- 

f  sr 

«£   o 

"-1  x 

Jj  cr 

'iif 

c  <•> 
""  u 

S  <n  0 

fc  «• 

!* 

§8. 

o  ^ 

01  JO 

Kind  of  Wire. 

feg 

i« 

.2s- 

^~S 

fjj 

S  ^ 

o  a, 

§  2 

•f  i 

—  '  i> 
By 

8  ^ 

.H  ^~ 

is 

"5.ti  3 

a 

5«° 

|| 

"S-< 

"83 

E9 

g.S 

1  j§ 

"^  <5 

5 

5"" 

5 

& 

5"* 

£ 

w~~ 

(S 

s*1 

56270 

28293 

33  *9 

28618000 

Annealed  wire 

^•3800 

g,   j 

•77 

53450 

28990 

22.0 

26997000 

Annealed  wire 

ii3° 

61500 

4jJCH_XJ 

03.  i 
75-° 



.76 

55880 

29758 

33-5 

27711000 

Annealed  wire 

•135 

61100 

39800 

49-4 

23000000 

•77 

53850 

29370 

33-3 

28718000 

Annealed  wire 

.136 

59500 

39200 

71.2 

25500000 

•77 

52770 

33722 

14.8 

27355000 

Annealed  wire 

45100 

35800 

76.8 

23500000 

•74 

52770 

28829 

33-5 

29273000 

Annealed  wire 

•  136 

59800 

34000 

72.7 

•77 

51320 

29294 

22    6 

28082000 

Annealed  wire 

•  J35 

62400 

•77 
•74 

« 
53778  , 
4888* 

27138 
28822 

25-4 
13  8 

28659000 
28137000 

Common  wire 
Common  wire 

.  no 
.109 

90900 

103000 

64000 

£.1 

30300000 
27500000 

•75 

49240 

28190 

14.0 

27520000 

Common  wire 

.  no 

104000 

60000 

51.0 

22900000 

•75 

50190 

30590 

17.8 

26237000 

Common  wire 

•"3 

93700 

68200 

60.5 

25200000 

•77 
•75 

51460 

47495 

29256 
30387 

22.4 

12.2 

25680000 
27613000 

Common  wire 
Common  wire 

.080 
.080 

113000 
113000 

45800 
56700 

41.9 
51.0 

27000000 
26500000 

48352 

30574 

17.3 

27177000 

Common  wire 

.079 

IT2OOO 

54300 

53-3 

26600000 

.76 

47I5I 

25982 

75-4 

21628000 

Common  wire 

•  079 

120000 

73600 

28.1 

26100000 

•77 

50351 

35720 

25-3 

27477000 

Common  wire 

.079 

IO9OOO 

54300 

40.4 

26400000 

•75 

48202 

28521 

14.7 

27888000 

Common  wire 

.080 

98300 

43-8 

27100000 

•75 

50703 

30558 

13.0 

23713000 

Annealed  wire 

.081 

99600 

61  .9 

26600000 

•75 

49223 

30517 

J5-2 

27126000 

Annealed  wire 

.082 

93500 

50400 

64-3 

•75 

49120 

29000 

17.8 

28290000 

Annealed  wire 

.082 

86300 

50400 

68.5 

27100000 

•75 

47060 

31700 

j  c    4 

Annealed  wire 

.082 

89900 

54OOO 

ej  •  7 

24900000 

:3 

47830 
51300 

29400 
26000 

17.8 

29.1 

29290000 
30100000 

Annealed  wire 
Annealed  wire 

.082 
.082 

97100 
93500 

57600 
39600 

55-0 

26100000 

•  76 

52400 

35000 

29.1 

25400000 

Annealed  wire 

.082 

71000 

50400 

67  .'i 

27000000 

:8 

53400 

52IOO 

29000 
26000 

24.9 
29.  i 

28200000 

Common  wire 
Annealed  wire 

.167 
.081 

57200 

45100 

65.6 
60.4 



CJTOO 

29000 

24.9 

Annealed  wire 

.082 

959 

553 

•76 

04-l*"K-' 

51500 

26500 

24.6 

33100000 

Common  wire 

.163 

935OO 
67400 

40100 

56-9 



•76 

52500 

242OO 

22    3 

Common  wire 

6l5OO 

.___- 

52.8 

•77 

77300 

34400 

26.5 

26800000 

Piano  wire, 

3 

•75 
•75 

53100 
52900 

31700 

24.9 
27.2 

27200000 
26100000 

No.  13 
Piano  wire, 

•  031 

345000 

29500000 

•76 

51600 

28700 

22  .  3 

26000000 

No,  23 

.048 

ofi^enn 

ononoooo 

4-ooX 

j                                    " 

1.  01 

40700 

14.4 



•76 

53100 

28700 

24.6 



.76 

52200 

33100 

26.9 

31000000 

•75 

50100 

31700 

22.6 

28700000 

•  76 

49400 

26500 

26.9 

.02 

50300 

31800 

16.9 

27200000 

.01 

47000 

32500 

20.6 

27700000 

.01 

50400 

30000 

25.8 

28300000 

.02 

49600 

31800 

23-9 

26500000 

.OI 

50200 

30000 

32-5 

26800000 

.02 

50500 

29400 

30.6 

26200000 

.01 

51400 

30000 

29.2 

28300000 

.02 

50400 



20.4 

28200000 

.02 

50200 

31800 

I5-1 

27200000 

.01 

48100 

30000 

32-5 

27700000 

.OI 

50600 

30000 

27-5 

25800000 

•77 

48600 

25800 

52.6 

28000000 

•74 

53900 

27900 

38.6 

29700000 

• 

•74 

54000 

25600 

18.0 

29700000 

76 

AI    8 

.70 

.76 

53500 

30900 

41.0 

33-4 

27600000 

TENSILE    TESTS   OF   WROUGHT  IRON. 


407 


In  Heft  IV  (1890)  of  the  Mitt.  d.  Materialpriifungsanstalt 
in  Zurich  is  an  account  of  a  set  of  tensile  tests  of  wrought-iron 
and  mild-steel  angles,  tees,  and  channels.  The  following  is  a 
summary  of  his  results  for  wrought-iron  shapes :  — 


ANGLE-IRONS. 


•a' 

rt 

g 

u 

•J  w"o 

•-  v-"5 

*j'u>S 

•SU 

Modulus  of 

5 

Dimensions, 
Inches. 

V 

a 

I|s 

X  g  3 

*ls 

(2-S  jj 

*O  3  2 

Ifc 
1* 

Elasticity, 
Pounds  per 
Square  Inch. 

3 

& 

J  o  a" 

13  o  o* 
j^CXyj 

T3  u 

Lbs. 

2 

2.76  X  2.76  X  0.31 

l6-53 

49910 

25020 

37680 

9-5 

28824000 

4 

2.76  X  2.76  X  0.51 

27.62 

49060 

20190 

32560 

II.7 

28070000 

6 

3-54  X  3-54  X  0.35 

26.21 

50620 

25310 

15.8 

28269000 

8 

3-l5  X  3-54  X  0.55 

38-51 

5II90 

25310 

32000 

l6.4 

27786000 

1O 

4.13  X  4'*3  X  0.47 

35.48 

49200 

28010 

33*30 

12.  O 

28537000 

12 

4.13  X4-13  Xo.67 

55-24 

46070 

22750 

32280 

10.2 

28554000 

M 

5.12  X  5.12  X  0.67 

62.09 

47780 

22610 

3043° 

12.  0 

27985000 

16 

5.12  X  5-12  X  0.87 

102.61 

48490 

31140 

12.3 

TEE-IRONS. 


3 

3".  60  X  3"  -35 

22.88 

52470 

25880 

3768o 

14.2 

27615000 

4 

" 

49200 

23610 

34700 

15-5 

27672000 

7 

3".  94  X  3"-94 

31.75 

51760 

21610 

38820 

11.7 

27857000 

8 

54040 

18630 

354*0 

21.3 

27743000 

9 

10 

5".  90  X  S'^94 

46.^91 

536lo 
52900 

23600 
22890 

36970 
3598o 

19.0 
14-5 

27402000 
28255000 

CHANNEL-IRONS. 


I 

4.13  X  2.56 

28.43 

50630 

23329 

35120 

15-9 

27544000 

a 

44 

49200 

24170 

33700 

12.7 

27885000 

4 

4.13  X  2.64 

31.65 

5432o 

23040 

36690 

20.6 

27772000 

5 

6.93  X  2.83 

48.89 

51760 

24460 

35550 

19.9 

27658000 

6 

"      " 

" 

54610 

23320 

34270 

17-5 

27999000 

7 
8 

6.93  X  2.^91 

S4ti43 

51900 
52900 

19620 
24320 

35690 
30860 

20.4 

14.5 

27487000 
29663000 

9 

8.46  X  3-35 

85.68 

52050 

22330 

3456o 

20.9 

27701000 

10 

4k         4*T 

'* 

5347° 

24170 

36260 

17.0 

28710000 

la 

8.46  X  3-50 

92.33 

52760 

32040 

34840 

ix.  9 

28568000 

408 


APPLIED   MECHANICS. 


TENSILE    TESTS    MADE   SUBSEQUENTLY   AT   THE   WATERTOWN 

ARSENAL. 

Here  will  next  be  given,  in  tabulated  form,  the  results  of  a 
number  of  tensile  tests  made  on  the  government  machine  at  the 
Watertown  Arsenal. 

The  following  tables  of  results  on  rolled  bars,  from  the  Elmira 
Rolling-Mill  Company  (mark  L)  and  from  the  Passaic  Rolling- 
Mills  (mark  S),  are  given  in  Executive  Document  12,  ^.Jth  Con- 
gress, 1st  session,  and  in  Executive  Document  /,  <ffth  Congress, 
2d  session. 

SINGLE   REFINED   BARS. 


1 

c 

0 

^ 

rt 
% 

Sectional  Area,  in 
square  inches. 

.£  U 

id 

J  8,  . 
•2  rf-5 
a  a  £ 

w 

Ultimate  Strength, 
in  Ibs.,  per 
Square  Inch. 

$> 

c 

'§*? 

S-5 
I'S 

Contraction  of 
Area,  %. 

Appearance  of 
Fracture. 

Modulus  of  Elas- 
ticity at  Load  of 
20000  Lbs.  per 
Square  Inch. 

1^ 
ta 

t* 

II 

L   i 

3.06 

28500 

52710 

18.4 

33-3 

95 

5 

26981450 

L     2 

3  06 

29500 

53630 

16.4 

36.0 

92 

8 

27826036 

L   3 

3.06 

29000 

52090 

21.4 

34.6 

95 

5 

28419182 

L   4 

3.06 

29000 

5  '440 

15.0 

20.3 

90 

IO 

30888030 

L   5 

6.46 

27500 

50500 

H-5 

27.6 

95 

5 

27826036 

L   6 

6.40 

27500 

50530 

17-3 

22.3 

70 

30 

27118644 

L   7 

6-39 

27000 

50200 

18.0 

22.5 

95 

5 

27444253 

L   8 

3-24 

- 

51667 

22.0 

36.0 

70 

3° 

28318584 

Round. 

L   9 

3.20 

- 

50844 

I6.3 

22.0 

'5 

85 

27972027 

u 

L   10 

3.20 

- 

53062 

2I.O 

4O.O 

95 

5 

28119507 

« 

S   ii 

3.08 

28500 

48640 

'3-3 

24-3 

100 

Slightly 

27586206 

S    12 

3.08 

28000 

50390 

16.9 

35-i 

100 

o 

27586206 

s  13 

3-05 

28500 

47050 

9.0 

22.0 

IOO 

o 

27874564 

S   iS 

6.40 

26000 

49700 

17.1 

19.2 

85 

15 

29906542 

S   16 

6.40 

24000 

49280 

15-7 

177 

85 

!5 

26490066 

S   17 

6.41 

24500 

48740 

14-3 

17.3 

80 

20 

28119507 

S   18 

3-i7 

24600 

49680 

19-5 

32.0 

IOO 

Slightly 

27972027 

Round. 

S   19 

3-17 

25870 

49338 

18.3 

38.0 

IOO 

0 

29357798 

" 

Cinder 

S   20 

3-i7 

24920 

48864 

18.4 

37-o 

IOO 

at  centre 

27729636 

DOUBLE   REFINED   BARS. 


409 


DOUBLE    REFINED    BARS. 


Mark  on  Bar. 

Sectional  Area,  in 
square  inches. 

•£  S 

~-  H, 

!  j? 

3  I  . 

u  -  -c 

llJ 

H 

Ultimate  Strength, 
in  Ibs.,  per 
Square  Inch. 

S 

g 

g^ 
11 

I  -s 
w 

Contraction  of 
Area,  %. 

Appearance  of 
Fracture. 

Modulus  of  Elas- 
ticity at  Load  of 
20000  Ibs.  per 
Square  Inch. 

1  ^ 
fa 

si  ^ 

\  i 

u  - 

L  20  1   3.06 

29000 

5356o  15-3 

37-9 

100 

o 

2763385  l 

L  2O2 

3-03 

30000 

52650  16.2 

20.6 

85 

15 

34042553 

L  203 

3.06 

32500 

53500  1  16.5 

27-5 

TOO 

0 

28169014 

L  204 

3.06 

32500 

54480  15.4 

24.8 

TOO 

o 

29090909 

L  205 

6-33 

27000 

51230  17.8 

24.2 

80 

20 

28119507 

L  206 

6-34 

27500 

50500  17.6 

21.  1 

100 

Slightly 

29629629 

L  207 

6-34 

27000 

51030  ;  21.4 

31-9 

100 

o 

27826086 

C^  up- 

L  208 

i 

3.20 

— 

50156  '22.7 

43° 

IOO 

shaped 

28021015 

Round. 

L  209 

3.20 

- 

49937  22.6 

45-o 

TOO 

" 

28622540 

" 

L   210 

3.20 

- 

50188  19.9 

43-o 

IOO 

" 

28985507 

" 

S   211 

3-05 

29500 

5II50  J22.0 

3i-5 

IOO 

o 

32989690 

S   212 

3-°5 

28500 

5IIIO   22.0 

36.1 

IOO 

o 

25559105 

S  213 

3-" 

29500 

51860   22-5 

39-2 

IOO 

o 

26446280 

S  215 

6.31 

27500 

50980 

19.1 

23.6 

95 

5 

29357798 

S  216 

6.38 

27OOO 

50770 

20.7 

29.6 

IOO 

0 

28268551 

S  217 

6-33 

27000 

51340 

19-3 

35-2 

IOO 

o 

28070175 

S  218 

3-i7 

24610 

50631 

20.4 

41.0 

IOO 

o 

28622540 

Round. 

Cup- 

S  219 

3-i7 

— 

50915 

25-5 

44.0 

IOO 

shaped 

28268551 

" 

S   220 

3-^7 

— 

50205 

23-7 

44.0 

IOO 

28070175 

The  moduli  of  elasticity  had  not  been  computed  in  the 
report,  but  have  been  computed  in  these  tables  from  the  elon- 
gations under  a  load  of  20000  Ibs.  per  square  inch  in  each  case, 
as  recorded  in  the  details  of  the  tests. 

In  these  reports  are  also  to  be  found  tensile  tests  of  iron 
from  other  companies,  as  the  Detroit  Bridge  Company,  the 
Phoenix  Company,  the  Pencoyd  Company,  etc.  Some  of  these 


4io 


APPLIED   MECHANICS. 


tests  were  made  to  determine  the  effect  of 

rest  upon  the  bar 

after  it  had  been    strained  to   its  ultimate 

strength,  also  to 

determine  the  strength  after 

annealing.     The  following  table 

shows  these  latter  results  : 

•o-o 

5 

8|  . 

a 

v  c-2 

& 

li! 

c          a         «       c     c 

rt            rt                       rt      rt 

1 

S     S             § 

o 

far                 hfl                 **           M        M 

hi) 

&        fci                        hli 

u 

»5^  .   (c"£          g.^  vC    i5C 

•        *        w 

iJC      "*C                  *C       <3 

fits 

3  R>*   3  8.  J           2    ^«  °  v 

&•*•!,:      § 

8*  vg^             &*  g 

p 

£  en  w    ^  «  °°    «^S  «     CB'  ">  g"  « 

•  gjftif  E&8  2«'2o 

0.0*5     0^2*3    iJ  2^    ^  3J31; 

-      10         wwrt          i    «« 

!i  III    il 

«&  ,?3.      .•  »•&  g 

2  £  2  S"        1  I  S"  ? 

ja"3  ja  g          ^  .0  3    a 

££     ££     £o     £    £ 

£     fc    0       Q  £ 

£___£          E  E     '£ 

•o  -d  'uopoas 

—  .  —     —  ,  — 

—  .  — 

B9JV  JO 

M         1            VO        CO          «0  M                •*          tx 

1      ^    S       S  co 

*      *         «  "     " 

•3  'd  'qiSuaq 
\vu\BuQ  cuoaj 
uoiiBxfuo^g 

M-^-Min^-O         coco 

MM          MM          HM             H          M 

ro       vo       co           N     sn 

C\                    O^                           VO         H                  M 

M                    H                            M         C1                 ti 

d  5ti« 

S>  8     2  vg    8>S\     8    (8 

MO            HONCOCO              »OOv 

8  5-    ft    &       5-  vS 

WVO        VO          H                 IN       10 

R.       §      8   8  v8      S 

O          S>        ij-    in    co        o 

O    (J          ""    S    «5    ri 

**    &« 

«n  vo       m  vo       invo         in      «n 

co  •*      in     vo           •*    in 

vo           in       vo     «n  vo        vo 

§  a      .H  --  v  c 

{/)  t»C       w  Q  P*H~I 

0               S     O        Q               O        Q 
o             o    ^      o             o       o 

»o    I       >n    •*•      0  I         m      m 

J.     8     8      |  § 

O             0                 Q     O 

«         S             o    o 

0>           Jo         f      in   oo           | 

o  s^lcfl 

I?        ^   S5    -S         ST    ^ 

«         «       in          «     N 

\ft          in              co    in 

jlta 

S1  .     ff  i     'Si      ^    i 

^0          vg       ^            =0^ 

CO                                  vo      tv           « 

rx           i          I      o    vo        vo 

w?-    "^cJf 

to            ro            ro           vo 

in        co      ei           N     co 

M                              ro    N          « 

*|      ^gd 

181*81 

ET      S     S       8.8 

jn          w               M     ^t*        w 

u  c       £  c""1 
H  «j        I"1 

M                       M                       M                       « 

" 

K*     5. 

T3  C 

S  ,     o"  .     S1!      5*    • 

R     ?  <?      <2  ? 

<^          ff         ,      S1*        o? 

m^_  fr>_^  °°        ^ 

•*•         ro      «            M     ro 

M               M                     ro     d            M 

Condition  of  Bar 
when  Tested. 

:p  11  :,   :  i 

•11  -al  -I    •  2 

•||  'l|   "S      '  § 

llgl^ll?  1  •s 
Hitiln  *  s 

O&<      OPS      Ooi      O    OS 

•og   '     '     -•a"  S   '     ' 

£&•  ;  I«J'  ; 

H.S  *   •   i-.s  *   • 

g-a  .     .     fc  s-o  .     . 

•si  .  _  r-3-_ 
il  •  4  nl  •  4 

««  o.h  .SP     w  «  c.h  .5P 

srtrto   «Ertrto 

/ 

1  !!!i  •  i  i 

=  »!ls  •  s  s 

s    2~s£     -s    -5 

73         *?,  «  T)  1-      c    -0         -0 

8    818558    S 

4)            OJCJDrt'CvD            OJ 

pi     D:    as     o  oi     rt 

i      g;    i          I 

s       R  **     R 

oo              oo'          P.  oo        oo" 
co                co           0s     co         co 

DOUBLE  REFINED  BARS. 


411 


Some  tests  were  made  to  determine  the  values  of  the 
modulus  of  elasticity  of  the  same  iron  for  tension  and  for 
compression ;  and  these  were  found  experimentally  to  be 
almost  identical,  as  was  to  be  expected.  For  these  tests  the 
student  is  referred  to  the  reports  themselves ;  and  only  cer- 
tain tests  on  eye-bars  of  the  Phcenix  Company  will  be 
appended  here. 


Arsenal 
Number. 

Outside 
Length, 
Inches. 

Gauged 
Length, 
Inches. 

Sectional 
Area, 
Sq.  In. 

Ultimate 
Strength, 
Pounds 
per  sq.  in. 

Contraction 
of  Area  at 
Fracture, 
per  cent. 

511 

67.75 

50 

1.478 

40600 

16.8 

513 

67.80 

50 

1.940 

39480 

13-9 

518 

96.05 

75 

5-103 

46720 

8.1 

Quite  a  number  of  tests  of  the  iron  of  different  American 
companies  are  to  be  found  in  the  "Report  on  the  Progress  of 
Work  on  the  Cincinnati  Southern  Railway,"  by  Thomas  D. 
Lovett,  Nov.  i,  1875. 

For  these  the  student  is  referred  to  the  report  named. 


WROUGHT-IRON    PLATE. 


The  following  table  contains  some  tests  of  wrought-iron 
plate  and  bars  made  on  the  Government  testing-machine  at 
Watertown  in  1883  and  1884  for  the  Supervising  Architect  at 
Washington,  D.C. 


412 


APPLIED   MECHANICS. 


§ 

.g  '3  0 

I"  Mi 

fi-i  e  .  i3  _e  fl  ^  *•* 

rf        11181  8 

§       s-s-sg.s«g 

£•*•»•*      •» 

.0-  -  -   w 


8s  §r 

"1    81 

to       J3  (fl 

£    EE    tt,          £ 


HiHit 

a;  ".n  ^-«    w  C  r*       **      T3  •* 

' 


a 


</-, 


6*OOc*)NOOfi 
fO         NNNflO 


gation 
acture. 


E 
t 


M   O   txO   O   rorotxd  "»•  ON  M 


O   O  vO   0  »  O  •*•  iO*O   O  00   •<*•    ir>  t^  t^\O  OOOOO 


•satpui  jqSp  uj 


-saqoui 
' 


Is 


Ultimat 
Strengt 


OO 

ir)M 

ro  CO  M  \o  -^  fO 
«  00  00  <0  00  M    0  000  ON  OOO  r^  ON  O  • 


,s,s, 


N?   O        O   0   O   0   0   O 
«o  &.o  «<?    5»«oo N  ?" 

cKoo  o  o  t-*  *o      in  co  ti  m  u">  ^- 


O  O 

S£S 

VJNO     ^"NO  NO 

CNlNNNCMNNNN 


0  o  oo  oo  10  m  1000  10  M     t~-  o  8  oo  m  ON  ON       •*•  N  o5  8  -<i-v 

ONON^'-^'i-'    ON»O»OfO-^     ON  ONOO    IO  »O  IO  ^        OO  NO  VO     1^*00    1 


p 


8OOOQOOOOOOQOOOOOOOOQOOOQO     OOOOQQQ-S    OOOOOO 
^5^0ooOO^om5>oOooNoi5»oo«oOOOvooOooooto  %•<%  00,,.  JoOOOtOM 

WNO    HI    0    •^•ONfOtNNO    IOOO  OO    Ht    ONrOCNl    CNN    M    M    CN1    lot-1    0    t^fO     O^W    CN1    txONONfONU    ^"MOO    ^-ONH 
t^  rxNO  NO    10  -<J-  ^-NO  NO    JO  t>.  fxNO    -*CNlC^m-^-*--»»-i-(l-<NOiOrOlO     MMWIO  10NO  NO  T3  00  NO    t^OO  NO  NO 


•B3JV 


•  c<>  r-.  »ooo  oo  •^•^•ONIOQ  O  rot^o  O  NOOOO  o  ^*oo  ^-  o  »ooo  ^J-  O  0  ^  0  *^~  ^oo  en  t^  o  t^  tx  ro  TO 

C    O    »l    t^MONlOlOON  ON  NO    N    H    ON  ONNO    IO  ON  ON  ON  ONOO  OOMQOH  ONONOOOHrOTj-*JOONOOt-<O 

•-•  io»O'^'lo^'^"'*p^'^'l^'loio^'^'frifOfrNro^-^*fOfOioio»oio  rororoioioiotONVS  lo^fiovoio^o 

o-o'do'dddddddddddddddddoddddd  ddo'ddddf-idddddd 


ddQdddo'ddQodoooooooo  oooooo 


O      H      O      M  M      ONOO    00 

o d  d  d      odd 


.    ^"2  "fL  n"*    ^^    m  l^vo1  8    (?  ON  ^00* 

COO«^OONvNNromrsOOONO>ON 


^SEgisI!?  S-EtifSS 

-'     «     .5(5     M     .     HM        d     d     d     d     ^     H 


•uauipadc; 


a  a  a  s  8  9  s 

NO   t>.OO  ONOMCS 


-^-  tovo  NO    tx  rwo  NO   _    N    m  ^  ^^      M    ro  •*  10  IONO  <O 

OU_)lJlj1_3(j(j 


voo^tx 


ro  (r>  (rN^-'^--it-'^-^-^--4-->»--^-^-ioioioio  IONO  NONONONO    F-i^K    1010  io  ir 

NONONONONONONONONONONONONONONOVONONONONONONONONOVONO       NONONONO 

WROUGHT-IRON    AND    STEEL    EYE-BARS. 

In  the  report  of  the  Government  tests  for  1886  is  given  the 
following  table  of  tensile  tests  of  wrought-iron  eye-bars.  The 
wrought-iron  ones  were  furnished  by  the  General  Manager  of 
the  Boston  and  Maine  Railroad,  and  the  steel  ones  by  the 
Chief  Engineer  for  the  American  Committee  of  the  Statue  of 
Liberty. 


CO M PRESS IV E   STRENGTH  OF    WROUGHT-IRON.        413 


WROUGHT- IRON  EYE-BARS. 


Dimensions. 

K 

Elongation. 

1 

O 

"*"*   0) 

01  in    . 

Fracture. 

|-s  , 

*| 

bcx 

i  ( 

*o 

J3S 
«l 

O  C  oj 

$    V    0> 

.§HH 

SHH 

T) 

u.S 

o 

^co 

^.^  § 

fill 

£ 

1 

"!»! 

W  g 

r=3 

if 

t^s 

"o 

1 

W3   4; 

ia? 

G 
_0 

Appearance. 

c2£ 

•8 

H 

He/2 

cw 

Ojj 

Isl 

-d^^c 

'x-2  « 

1 

J 

5 

H 

2 

H 

c 

J 

S 

S 

3 

Ins. 

Ins. 

Ins. 

Lbs. 

Lbs. 

% 

% 

% 

Lbs. 

Lbs. 

238.55l5.oo 

1.14 

22456 

45105 

II  .7 

ii  .6 

31  .2 

28037000 

52763 

Stem 

Fibrous,     traces 

of  granulation. 

238.60 

5.00 

1.15 

22610 

44540 

9.4 

9.4 

29.6 

28125000 

50588 

" 

Fibrous,      70% 

Granular,   30% 

238.57 

4-99 

1.14 

21790 

43320 

7-8 

3.0 

26.4 

27950000 

48492 

1  ' 

Fibrous,      70% 

Granular,    30% 

238.64 

5.00 

1.  16 

22410 

39550 

5  •  i 

4-8 

9.8 

27355000 

54013 

1  ' 

Fibrous,      70% 

Granular,   30% 

238.62 

6.05 

i  .44 

19750 

43260 

12.05 

I  2  .06 

24.8 

28800000 

43166 

1  ' 

Granular,   80% 

Fibrous,      20% 

238.62 

6.05 

i  .44 

22730 

42020 

6.5 

6.6 

19.  2 

28301000 

41929 

" 

Granular  5%  at 

one      edge,       fi- 

brous   for     bal- 

ance of  fracture. 

The  gauged  length  of  the  bars  was  180  inches.  The  moduli 
of  elasticity  computed  between  5000  and  10,000  pounds  per 
square  inch. 

COMPRESSIVE    STRENGTH    OF    WROUGHT-IRON. 

In  regard  to  the  compressive  strength  of  wrought-iron,  we 
may  wish  to  study  it  with  reference  to — 

1°.  The  strength  of  wrought-iron  columns; 

2°.  The  strength  of  wrought-iron  beams  ; 

3°.  The  effects  of  a  crushing  force  upon  small  pieces  not 
laterally  supported  ; 

4°.  The  effects  of  a  crushing  force  upon  small  pieces  laterally 
supported. 

i°.  In  this  case  it  may  be  said  that,  by  reference  to  the 
tests  of  wrought-iron  bridge  columns,  the  compressive  strength 
per  square  inch  of  wrought-iron  in  masses  of  such  sizes  is  given 
by  the  tests  of  the  shorter  lengths  of  such  columns,  i.e.,  by 
those  that  are  short  enough  not  to  acquire,  when  the  maximum 
load  is  just  reached,  a  deflection  sufficient  to  throw  any  appreci- 
ably greater  stress  per  square  inch  on  any  part  of  the  column  in 
consequence  of  the  eccentricity  of  the  load  due  to  the  deflec- 


4H  APPLIED   MECHANICS. 

tion.  The  results  thus  obtained  are  naturally  lower  than  we 
should  expect  to  obtain  in  smaller  masses. 

2°.  In  this  case  the  evidence  that  there  is  goes  to  show  that 
the  compressive  strength  is  the  same  as  in  the  case  of  i°,  and 
hence  that  it  is  less  than  the  tensile  strength.  Indeed,  the  results 
of  tests  of  full-size  beams  show  a  modulus  of  rupture  greater 
than  the  compressive  strength,  less  than  the  tensile  strength  in  I 
sections,  and  greater  in  circular  sections;  all  this  being  what 
would  naturally  be  expected. 

3°.  If  a  small  cylinder  of  ductile  wrought-iron  is  tested  with- 
out lateral .  support,  and  with  flat  ends,  the  friction  of  the  ends 
against  the  platforms  of  the  testing-machine  comes  in  to  interfere 
with  the  flow  of  the  metal;  and  if,  besides  this,  the  ratio  of  length 
to  diameter  is  so  small  as  to  prevent  buckling,  then  the  specimen 
will  gradually  flatten  out,  and  it  becomes  impossible  to  find  any 
maximum  load,  because  the  area  of  the  central  part  is  constantly 
increasing. 

4°.  In  this  case  the  crushing  strength  per  square  inch  that 
causes  continuous  flow,  and  also  the  maximum  strength  per 
square  inch,  is  greater  than  that  where  the  specimen  has  no 
lateral  support.  Hence  follows,  that  in  the  case  of  wrought-iron 
rivets  it  is  .entirely  safe  to  allow  a  bearing  pressure  in  the  neighbor- 
hood of  90,000  or  100,000  pounds  per  square  inch,  according  to 
circumstances. 

§  223.  Wrought-iron  Columns. — Until  after  about  1880 
there  was  but  little  experimental  knowledge  on  this  subject  beyond 
the  experiments  of  Hodgkinson,  which  have  furnished  the  con- 
stants for  Hodgkinson's,  and  also  for  Gordon's  formula,  as  already 
given  in  §  208  and  §  209. 

These  formula  have  been  in  very  general  use,  and  it  is  only 
of  late  years  that  we  have  been  able  to  test  their  accuracy  by 
tests  on  full-size  wrought-iron  columns.  The  disagreement 
of  the  formulae  already  referred  to,  with  the  results  of  the  tests, 
has  led  to  the  proposal  of  a  large  number  of  similar  formulae, 


WRO  UGHT-IRON   COL  UMNS.  4  1  5 

each  having  its  constants  determined  to  suit  a  certain  definite 
set  of  tests,  and  hence  all  these  formulae  thus  proposed,  which 
are,  of  course,  empirical,  and  can  only  be  applied  with  safety 
within  the  range  of  the  cases  experimented  upon. 

A  few  of  these  will  now  be  enumerated;  and  then  will  follow 
tables  of  the  actual  tests,  which  furnish  the  best  means  of  deter- 
mining the  strength  of  these  columns  ;  and  it  would  appear  that 
it  is  these  tables  themselves  which  the  engineer  would  wish  to  use 
in  designing  any  structure. 

On  the  1  5th  of  June,  1881,  Mr.  Clark,  of  the  firm  of  Clark, 
Reeves  &  Co.,  presented  to  the  American  Society  of  Civil  Engineers 
a  report  of  a  number  of  tests  on  full-size  Phoenix  columns,  made 
for  them  at  the  Watertown  Arsenal,  together  with  a  comparison 
of  the  actual  breaking-weights  with  those  which  would  have 
been  obtained  by  using  the  common  form  of  Gordon's  formula 
for  wrought-iron.  The  table  is  shown  on  page  416. 

The  very  considerable  disagreement  between  the  breaking- 
loads  as  calculated  by  Gordon's  formula,  and  the  actual  break- 
ing-loads, led  a  number  of  people  to  propose  empirical  formulae 
of  one  form  or  another  which  should  represent  this  set  of 
tests,  and  also  others  which  should  represent  some  other  tests 
on  full-size  bridge  columns,  which  had  been  previously  made 
in  other  places. 

Of  these  I  shall  only  give  those  proposed  by  Mr.  Theodore 
Cooper,  which  are  as  follows  :  — 

p  f 

For  square-ended  columns  .     .     .    -j  =  —  -  -n  -  ry 

'  +   -*>) 


18000 


P  f 

For  phi-ended  columns  .    .    .    .    -j  = 


7.8000 


416 


APPLIED    MECHANICS. 


frlil 

- 


vO  ONfOONi^toior^>-  «  O  vO  VO  r^»  M  -TJ-  N  t 
Tj-vo-.   «   rorfLOO\O   O^OvOOO   -^-M    N   ro 
M«d-Of«COOOM-it»s  t^O  OO  OO   O   O   ro  «  « 


1    1    1    1 


tf  r^  r 


C\  O\<->   O  w   N 


^ 

w    1-1    M    O    ""If 

«J~>  rj-  >-o  i-o  LO  Tj 


O  Q  O  O 

?&S°2 

t~^  r^.vo  n 

lO  to  to  ^j- 


oo  ^o 
vo  vO 


CJ  "^  N         O   O   O   ^" 

•^  »-O  ^         CO  ^OSO    10 

ro  ro  to      ro  to  to  to 


10  II 


fOVO  OO    fO  N          O    ON 

^"  fO  O\  M    i     i    ^   i    M   O    i     i 

N    N   S   M    1      1    i-i     1    w   M    1      » 


N  fOM  oo  r> 

•        "  l\iJ     TfJUNMll^lMtJi  HNM     t^VO 

lNMMM>-iM'l»-ili-it-cl  'O  O    O    1-1    p 

dddddd          d      do  d  dddd 


ONOO     •VOVO>-O|fON'-iOSON|       1     to    I     fO  N  OO'-i'^f 

HtM'MMM'MMMlOO'        '      O      '     'O    O  Opl-lp 

do      odd      ddddd          d      do         dddd 


ioo  too  r^*"  MOO  Q  Ooo 

^NeiNcivi'fiM'cIci  MM  oo  06 


|_      M      O      O     ON  I--   I^O  VO 


•^•Tj-tOfOfOtOt) 


to  O 
O   ""> 


UOOOO 


N  ON  ONVO  vO  totoo  O  r^ 


r^oo  ON  o  M  N  to  rf  mvo  t^oo 


WROUGHT-IRON   COLUMNS. 


417 


418 


APPLLED    MECHANICS. 


And  he  gives,  for  the  values  off, 

For  Phoenix  columns /*=  36000; 

"  American  Company's  columns  .     .     .     ./"=  30000; 
"   box  and  open  columns /=  31000. 

He  deduces  these  values  of  f  from  some  tests  made  in  1875 
by  Mr.  Bouscaren,  combined  with  those,  already  referred  to, 
made  at  the  Watertown  Arsenal.  The  box  and  open  columns 
were  made  of  channel-bars  and  latticing.  The  tables  or  dia- 
grams presented  to  justify  the  formulae  proposed  can  be  found 
in  the  "Transactions  of  the  American  Society  of  Civil  Engineers  " 
for  1882. 

Besides  the  above  there  will  be  given  here  tables  of  three 
sets  of  tests  of  full-size  wrought-iron  columns,  viz. : 

i°.  The  series  made  at  Watertown  Arsenal,  this  being  the 
most  complete  set  of  tests  of  full-size  wrought-iron  columns  in 
existence. 

2°.  A  series  of  tests  of  Z-bar  columns  made  by  Mr.  C,  L. 
Strobel. 

3°.  A  few  tests  made  at  the  Mass.  Institute  of  Technology. 
Reference  will  also  be  made  to  the  tests  of  Mr.  G.  Bouscaren, 
and  to  those  made  by  Prof.  Tetmajer,  at  the  Materialprufungs- 
anstalt  in  Zurich. 

Graphical  rep  esentations,  however,  will  first  be  given  of  the 


10000 


results  of  those  tested  at  Watertown  Arsenal,  with  the  correspond- 
ing curves,  representing  (a)  the  formulae  of  Prof.  Sondericker  (see 


WROUGHT-IRON   COLUMNS.  419 

page  417),  and   (b)  that  of  Mr.  Strobel  (see  page  418).    These 
diagrams  will  be  preceded  by  the  corresponding  formulae. 

A  perusal  of  them  will  show  that,  for  values  of  —  less  than 

P 

a  certain  quantity,  which  Mr.  Strobel  assumes  as  90,  arid  Prof. 
Sondericker  as  80  for  flat-ended,  and  60  for  pin-ended  columns; 

p 

the  value  of  • —  (i.e.,  the  breaking-load    divided    by  the  area)  is 
A 

I  P 

constant.     For  greater  values  of  —  the  value  of  —  decreases,  and 

P  A 

for  this  portion  of  the  curve,  Prof.  Sondericker's  formulae  are  as 

follows : 

For  flat-ended  Phcenix  columns  he  recommends  Cooper's 
formula. 

For  lattice  columns  with  pin-ends,  reported  in  Exec.  Doc. 
12,  47th  Congress,  ist  session,  and  Exec.  Doc.  5,  48th  Congress, 
ist  session,  he  recommends  the  formula 

P_  =        34000 

A  = 

— 60 


12000 

For  the  box  and  solid  web  columns  reported  in  Exec.  Doc.  5, 
48th  Congress,  ist  session,  and  Exec.  Doc.  35,  49th  Congress, 
ist  session,  taken  together  with  Bouscaren's  results  on  box  and 
on  American  Bridge  Company's  columns,  he  recommends 

P  33000 

For  flat-ends -r= — - 

A. 


/       \2' 

-80 
p      / 


10000 


p  31000 

For  pin-ends  ..........  -;-= 


6000 


420  APPLIED    MECHANICS. 

In  these  formulae  P  =  breaking-load  in  pounds,  A=  sectional 
area  in  square  inches,  /  =  length  in  inches,  and  p  =  least  radius 
of  gyration  of  section  in  inches. 

Moreover,  the  numerator  in  each  of  these  formulae  is  the 

*P  / 

value  of  -j-  corresponding  to  the  case  when  —  is  less  than  80  in 
A  p 

flat-ended,  and  less  than  60  in  pin-ended  columns. 

Instead  of  the  above  Mr.  Strobel  recommends  for  value  of 

P  I 

-r  when  -  is  less  than  90,  35000  pounds  per  square  inch,  and, 
A  P 

for  values  of  —  greater  than  90,  the  formula 


r=46ooo-i25-. 

Moreover,  if  P'=safe  load,  in  pounds,  he  recommends 

/  P' 

(a)For-<90,         -^-  =  8000; 
p  A. 

I  P'  I 

(b)  For  —  >  90,         —r  =  1 0600  —  30—. 
p  A  p 

While  Gordon's  formula,  or  a  modification  of  it,  is  still  in 
use  in  many  bridge  specifications,  quite  a  number  of  them  have 
substituted  the  Strobel  formula,  or  a  modification  of  it. 

WROUGHT-IRON  COLUMNS  SUBJECTED  TO  ECCENTRIC  LOAD. 

All  the  formulae  given  thus  far  for  the  breaking  or  for  the 
safe  load  on  wrought-iron  columns  are  only  applicable  when 
the  load  is  so  applied  that  its  resultant  acts  along  the  axis  of  the 
column,  and  either  the  diagrams  on  pages  417  and  418,  or  the 
corresponding  formulae,  give  us  the  breaking-strength  per  square 
inch,  i.e.,  the  number  of  pounds  per  square  inch  which,  multiplied 


WROUGHT-IRON   COLUMNS.  421 


by  the  area  in  square  inches,  gives  the  breaking-load  of  the  column; 
the  safe  load  per  square  inch  being  obtained  by  dividing  the 
breaking-load  per  square  inch  by  a  suitable  factor  of  safety.  On 
the  other  hand,  whenever  the  resultant  of  the  load  on  the  column 
does  not  act  along  the  axis  of  the  column,  we  must  determine  the 
fibre-stress  due  to  the  direct  load,  and  to  this  add  the  greatest 
fibre  stress  due  to  the  bending-moment,  the  sum  of  the  two  being 
the  actual  greatest  fibre  stress,  and  the  column  must  be  so  pro- 
portioned that  this  greatest  fibre  stress  shall  not  exceed  the  safe 
strength  per  square  inch,  as  determined  by  dividing  the  breaking- 
strength  per  square  inch  by  the  proper  factor  of  safety;  and  this 
proceeding  should  be  followed  whatever  be  the  cause  of  the 
eccentric  load — whether  it  be  due  to  the  beams  supported  by  the 
column  on  one  side  being  more  heavily  loaded  than  those  on  the 
other,  whether  it  be  due  to  the  load  transmitted  from  the  columns 
above  being  eccentric,  whether  it  be  due  to  the  mode  of  connection 
of  the  column  to  the  other  parts  of  the  structure,  whether  it  be 
due  to  poor  fitting,  or  to  any  other  cause. 

TESTS   OF   FULL-SIZE   WROUGHT-IRON   COLUMNS. 

The  tests  made  at  the  Watertown  Arsenal  will  next  be  given, 
together  with  cuts  showing  the  form  of  the  columns ;  these  being 
taken  from  the  Tests  of  Metals  for  1881,  1882,  1883,  1884,  and 
1885. 

The  following  tables  are  taken  from  the  volume  for  1881.: 


422 


APPLIED    MECHANICS. 


jj 

1    1    I 

•6 

|= 

^0 

1 

rQ  „ 

it,  ..,..,;,,  ,|" 

1      1      1 

s        4        s 

"4}  3   .3    3      C 

o 

13    C 

b  ° 

rt    sj 
D^  O 

c             §5  »  «  » 

a    i    . 

0> 
•5          3  JS            •  N    t^OO    O    Tj-\O  OO  OO    M 

1  £~ 

^^gcgasN^^vSaa^ss0^ 

*-o  r^*  ^^  I-H  Q\  to  o  fO  toco  oo  r^  ONOO  c^ 

2       "«'             OQOOOOOOO 

C           3             u)OOOQOOOO'-'VOLr>1-r 
'£**•?          *o  T^-  i-o  o   ^-  o\\O   ^O  ro  ^^ 

OOOOQOOOOOOOOOO 
vn\O   ^O  O  "XXD   t^O   ^OLON    rONcO    f"O 

u  W) 
o  C 
[-•OOOOOOOOOOOOOOOOOO 

000000000000000000000000000000 

J.&    •  

"rt                      .OOOOOOOOO 

Is      .s^ic^s^^ 

Iooooooooooooooo 
VO  vO   ^vO  «O  t^'O  "-1   fOCTNt^O   ro  M   O 

F     -^ 

^•^•Tj-Tj-^TrTj-TJ-^TtTf^Tt^r^ 

§                 S-                            M    M    N    C4   vs- 

OVOVOVOOVOVOOOVOVOOO^^- 

«"w|3           g-vOvOvOvOvOVDvOvOvO 

VOVOVOVOVOVOVOVOVOVOVOVOVOOOOO 

"§3 

'    *  <u         

.      .    fi  -        

•d                       I* 

2              .   ."o-    

Is    -      -S, 

S    £s  g  

tS.    g.    c 

E    O"  Sa 

*o  *:                  ON  Q  ""^O  t^oO  M   N  1-1 

d£          1^88^22       ff 

N    ON  O    txOO    ONO"WO«COON»-'N 
rOMrOMMi-iNNNNNiHMMM 

J5                                          M     M     «      «     M     M 

WROUGHT-IRON  COLUMNS. 


423 


I         1 


11*9 


I. 

I 


£u   3   £ 


I...........I      I".  §s  i 

5  i««««i    jg  i 


O     '1 

•c     i 


<U3    S    3    3    A 
Q  U 


ll 


fO  ^O  fO  ro  CO  ^O 


000000000000000000000000 

»—  (  ^  c^   ^~  ^ovo  CTvoo  ^»  O  *^   O  ^   ^^  O   ""<  f^  ^O  ^)  ^   Q   O   O  ^O 


^-Ti- 

N    M 


.ooooooooooogooovooooooooooo^ooo 

c  oo  oo  ^o  »-  o\  t^oo  "^"-1  Ooo  o\"">i-<  ^o  Tt-g  o  r^  mvo  N  i^oo  -^f  •<*•  «  oo  • 
""  ^  -^  ^o  10  c\"O  r^.  r^oo  oo  vO  V>VI\Q  1^.00  OQfOvr>>-iO'^T*'OSfO'-iON^1-^ 


H  OO  OO  O  O 


OOVOVOOOOOOOOOC^MOOOOOOOOOO 


eOOOOOOOOOOOOOOOOOOOOOOOOOOOOOONNNNNMNNMN 


424 


APPLIED    MECHANICS. 


•a 

1s 

.S 


fe      Q 


^. 

pi 

.S*£ 

T3  "3  "3 
«   0   0 


72   rt^   rt^  So-    - 

g£gsgs  S 

IsJsJ  *    -H 

•g  1,,  i 

3:     =     3     S     3  *                 IS 


II 


= 
- 


i 


^OOQQOOOOOOQOO 
O\  r^  O  VO  "O   TJ-OO   w00^*«fie»w 


OOOOOOOOOO 
fn  ""KQ  -^  TJ-  ON  vo  rf  vn  -^f 
O  O  O  N  n  u->  ro  O  «  Q 


C3        j§  <«  ^ooo  O\  "•>  ^  -^-  <*S  r~«vO  ^O  fO>-ivo  M  •'J'O'Ooo  "^w  r»  coro  t^.oo  S  **>»•• 

g  ^  <  00001^1^        VO    ^    £>£>£>2    «"£    «"£    J^^'S'S    W    £    N    N    N    N    S 

N 

«n 

C  fO  f">  t^  fO  fO  t-^00  to  "-)  ro  fO  ro  ro  fOOO  OO  VO  •«*•  -fvo  MD  r>>i>.tON  M\OON 
***  co  ro  to  w  N  ^O  fOoO  OO  t^  r^x  t^»  r^  i^  i^  i^*  r^  O  O  t*x  t^  ON  ON  ON  O  O  ON  ^ 
^  °5  crNNNNWc5cirof'5fOfOfOfOfO'<i-'^-'<floiOTi-Ti-ioio  iovd  NO  *ovd 

—  "^ 

< 
« 

x'  c  •-•  \O  vO  t^«»  t^»  fO  rOOO  OO  00  f^»  ^5  ^^  ONOO   O  O  t^  ^O  fO  ONOO   w   N  t*>.  fO  to  ONOO 

g  _    _    M    N    rj-^  MNNNT}-i-iH<>-iWNN'<$-i-ii-il-iNNN-'*- 

< 

EG 

u 

*H  .Sj 

o         ^ 

c/, 

g 

w 

H 

§    -      : 

a" 

^  V. 

g  .  M  •*$•  *r\t~*.  t<£  t*»VD  *O'~       "~        -     -  — 

O  |S 


WROUGHT-IRON   COLUMNS. 


42  s 


CX  O    &,  O 
SJ.C   3  .£3 


•  O  vo  *"O  o  co  O  f^ 

52  O   ^   r^  O  ON  O   ON 

^  10  10  i^\o  ON  N  oo 

(4  M  ft  ti  N  <*)(* 


. 

•y  r^  i-<  vo  M 


»o  ON  ON  "">  >-o 
r^^.  «  O^ON 
fC.  ON  O\  ^  N 


.BOOOOOOOO 


rfvO  \OOOOO  O  O  N  N 


«    5    5    S    5    5    S 


a  | 

H     J5 


bo 

if  1 


S    | 

^  S 


Jl 


I    If 

1   I 
1  I 


00*0 


00 

oo  N 


"™ 


-oooooooooooooo 


S     3     3     S     S     S 


J3s    3    3    3    S    3 
E 


426 


APPLIED   MECHANICS. 


'rf 

IS 


rS 

.   C 
*O   O 

QJ     N 

3'S 


.s'S'S 


6  o 


g 

t 

3 

0 

• 

a 

u 

.9-j 

Manner  of  ' 

t  Deflected  ho 
(  and  upwar 

1 

o 

• 

|je 

| 

t/3   c 

~   S 

Ultimate 

1 

3 

4 

Ij 

.s^ 

°<t! 
c^^ 

"  ~ 

-C 

.5  °° 

§ 

J 

«   N 

o  g" 

-s 

• 

c 

ID 

.s 

a 

o 

"rt 

V 

C 

0 

^^ 

& 

it- 

WR 0  UGHT-IR  ON  COL  UMNS. 


427 


428 


APPLIED   MECHANICS. 


WROUGHT-IRON  COLUMNS. 


429 


The  next  table  taken  from  the  volume  for  1882  men- 
tioned above  contains  the  results  of  some  compressive  tests 
of  wrought-iron  I-beams  placed  in  the  machine  with  the  ends 
vertical  and  tested  with  flat-ends  ;  also  of  some  tensile  speci- 
mens cut  off  from  two  of  them. 

TESTS   OF   I-BEAMS   BY   COMPRESSION. 


Width 
of 

Thick- 

Total 

*J 

•C 

tx 

Sectional. 

Ultimate  Strength. 

Length. 

Flange. 

Web. 

Depth. 

(U 

Area. 

Actual. 

PerSq.  In 

In. 

In. 

In. 

In. 

L  s. 

Sq.  In. 

Lbs. 

Lbs. 

I 

57.06 

5-45 

0.64 

9.00 

228 

14.40 

545  TOO 

37854 

2 

155-45 

4.40 

0.40 

10.52 

443 

IO.26 

207000 

20170 

3 

191.90 

3-56 

0.40 

9.08 

365 

6.85 

85380 

12460 

4 

191.90 

3-59 

0-43 

9.09 

38i 

7-15 

85200 

11916 

5 

119-85 

2.98 

0.28 

6.  ii 

139 

4.18 

IOI200 

24210 

6 

i8o.33 

3.60 

0.42 

6.96 

303 

6.05 

84650 

13990 

7 

192.04 

3.58 

0-45 

7-94 

355 

6.65 

83400 

12540 

8 

192.90 

3.60 

0.44 

7.98 

353 

6-59 

92300 

I40IO 

9 

215-88 

4.28 

0.40 

10.52 

561 

9-30 

I49OOO 

I6O2O 

10 

264.08 

4-49 

0.48 

10-53 

747 

10.19 

II3IOO 

IIIOO 

ii 

264.08 

4-43 

0.50 

10.51 

767 

10.46 

107800 

10306 

12 

264.00 

4.90 

0.53 

I5-I5 

1085 

14.80 

184700 

I24OO 

13 

263.95 

4.84 

0-53 

14.74 

1081 

14.74 

I87OOO 

12686 

TESTS   OF  SPECIMENS   FROM   NOS.    I    AND   2   BY  TENSION. 


Cut    from 
Flange 
or  Web. 

Width. 
In. 

Depth. 
In. 

Sectional 
Area. 

Sq.  In. 

Ultimate  Strength. 

Contrac- 
tion of 
Area. 

Per  Cent. 

Actual. 
Lbs. 

Per  Sq.  In. 
Lbs. 

f 

Web. 

3-00 

0.65 

1-95 

103300 

52970 

IO 

Web. 

3-00 

0.50 

I-5I 

65400 

43340 

3-9 

1 

Flange. 

4.00 

0.75 

3-01 

146400 

48640 

19.6 

1 

Flange. 

4.00 

0.76 

3-02 

I47IOO 

48640 

15-9 

f 

Flange. 

3.00 

0.51 

1-53 

55400 

36210 

n.  I 

I 

Web. 

3.00 

0.40 

I.I9 

52900 

44640 

16.5 

43°  APPLIED    MECHANICS. 

Next  will  be  given  the  set  of  tests  which  is  reported  in  the 
volumes  for  1883  and  1884. 

The  following  is  quoted  from  the  first  of  the  two  : 


"  COMPRESSION    TESTS   OF   WROUGHT-IRON  COLUMNS,   LATTICED,  BOX, 

AND    SOLID    WEB. 

"  This  series  of  tests  comprises  seventy-four  columns,  forty 
of  the  number  having  been  tested,  the  results  of  which  are 
herewith  presented. 

"The  columns  were  made  by  the  Detroit  Bridge  and  Iron 
Company. 

"  The  styles  of  posts  represented  are  those  composed  of — 

"  Channel-bars  with  solid  webs  ; 

"  Channel-bars  and  plates  ; 

"  Plates  and  angles  ; 

"  Channel-bars  latticed,  with  straight  and  swelled  sides ; 

"  Channel-bars,  latticed  on  one  side,  and  with  continuous 
plate  on  one  side. 

"  All  the  posts  were  tested  with  3^-inch  pins  placed  in  the 
centre  of  gravity  of  cross-section  ;  except  two  posts  of  set  Ny 
which  had  the  pins  in  the  centre  of  gravity  of  the  channel- 
bars. 

"  This  gave  an  eccentric  loading  for  these  columns,  on  ac- 
count of  the  continuous  plate  on  one  side  of  the  channel- 
bars. 

"  The  pins  were  used  in  a  vertical  position,  unless  other- 
wise stated  in  the  details  of  the  tests. 

"  In  the  testing-machine  the  posts  occupied  a  horizontal 
position. 

"  They  were  counterweighted  at  the  middle. 

"  Cast-iron  bolsters  for  pin-seats  were  used  between  the  ends 


WROUGHT-IRON   COLUMNS.  43! 

of  the  columns  and  the  flat  compression  platforms  of  the  test- 
ing-machine. 

"  The  sectional  areas  were  obtained  from  the  weights  of  the 
channel-bars,  angles,  and  plates,  which  were  weighed  before 
any  holes  were  punched,  calling  the  sectional  area,  in  square 
inches,  one-tenth  the  weight  in  pounds  per  yard  of  the  iron. 

"  Compressions  and  sets  were  measured  within  the  gauged 
length  by  a  screw  micrometer. 

"  The  gauged  length  covered  the  middle  portion  of  the 
post,  and  was  taken  along  the  centre  line  of  the  upper  chan- 
nel-bar or  plate,  always  using  a  length  shorter  than  the  dis- 
tance between  the  eye-plates,  to  obtain  gaugings  unaffected  by 
the  concentration  of  the  load  at  those  points. 

"  The  deflections  were  measured  at  the  middle  of  the  post. 
The  pointer,  moving  over  the  face  of  a  dial,  indicated  the 
amount  and  direction  of  the  deflection. 

"  Loads  were  gradually  applied,  measuring  the  compressions 
and  deflections  after  each  increment ;  returning  at  intervals  to 
the  initial  load  to  determine  the  sets. 

"  The  maximum  load  the  column  was  capable  of  sustaining 
was  recorded  as  the  ultimate  strength,  although,  previous  to 
reaching  this  load,  considerable  distortion  may  have  been  pro- 
duced. 

"  Observations  were  made  on  the  behavior  of  the  posts 
after  passing  the  maximum  load,  while  the  pressure  was  fall- 
ing, showing,  in  some  cases,  a  tendency  to  deflect  with  a  sudden 
spring,  accompanied  by  serious  loss  of  strength. 

"  The  slips  of  the  eye-plates  along  the  continuous  plates 
and  channel-bars  during  the  tests  were  measured  for  certain 
posts  in  sets  F,  G,  H,  and  /.  The  measurements  of  slip  were 
taken  in  a  length  of  10  inches  or  20  inches,  one  end  of  the 
micrometer  being  secured  to  the  eye-plate,  and  one  end  to  the 
channel-bar.  The  readings  include  both  the  compression 
movement  of  the  material  and  the  slip  of  the  plates. 


43 2  APPLIED    MECHANICS. 

"  Columns  H,  7,  Z,  and  J/ were  provided  with  pin-holes  for 
placing  the  pins  either  parallel  or  perpendicular  to  the  webs  of 
the  channel-bars. 

"  After  the  ultimate  strength  had  been  determined  with  the 
pins  in  their  first  position,  a  supplementary  test  was  made,  if 
the  condition  of  the  column  justified  it,  with  the  pins  at  right 
angles  to  their  former  position  ;  thus  changing  the  moment  of 
inertia  of  the  cross-section,  taken  about  the  pin  as  an  axis. 

"  The  experiments  with  columns  N  show  how  much  strength 
is  saved  by  employing  pins  in  the  centre  of  gravity  of  the  cross- 
section.  Where  such  was  not  the  case,  the  columns  showed 
the  effect  of  the  eccentric  loading  by  deflections  perpendicular 
to  the  axis  of  the  pins,  from  the  initial  loads,  which  resulted  in 
their  early  failure." 


WKO  UGHT-1RON   COL  UMNS. 


433 


TABULATION   OF   EXPERIMENTS   ON    WROUGHT-IRON   COLUMNS 
WITH    3J-INCH    PIN-ENDS. 


Ultimate 

Length, 

Strength. 

Centre 

Sec- 

No. of 
Test. 

Style  of  Column. 

to 
Centre 

tional 
Area. 

Total, 

Lbs. 

Manner  of  Failure. 

of  Pins. 

Lbs. 

per 

In. 

Sq.  In. 

Sq.  In. 

Set! 

A. 

752 

u 

_J 

126.20 

9-831 

297100 

30220 

Deflected    perpendicular 
to  axis  of  pins. 

757 

*•-„     " 

120.07 

10.199 

320000 

31380 

Sheared     rivets    in   eye- 

if 

'           t;  1 

plates. 

755 

X> 

10 

180.00 

9-977 

251000 

25160 

Deflected    perpendicular 
to  axis  of  pins. 

756 

r— 

~~n~\ 

180.00 

9-977 

210000 

21050 

Do.            do. 

753 

*~~z 

•6—  r> 

240.00 

9-732 

188600 

19380 

Do.             do. 

754 

i 

240.10 

9.762 

158300 

16220 

Do.            do. 

i 

SetlD, 

1642 

240.00 
240.00 

16.077 
16.281 

425000 
367000 

26430 
22540 

Deflected    perpendicular 
to  axis  of  pins. 
Do.            do. 

~] 

r=! 

1646 

n? 

320.00 

16.179 

3I8800 

19700 

Do.            do. 

1647 

\ 

320.10 

16.141 

283600 

Do.            do. 

Hj,    ' 
8  > 

.   Set 

a 

,653 

*/8 

I 

320.00 

17.898 

474000 

26480 

Deflected    perpendicular 
to  axis  of  pins. 

1654 

i 

320.00 

19.417 

49IOOO 

25290 

Do.            do. 

Le^ 

1             1 

Setjp. 

,645 

e  —  s' 

Tr 

319.95 

16.168 

453000 

28020 

Deflected  parallel  to  axis 

of  pins. 

1*50 

"*'" 

IL 

320.00 

16.267 

454000 

27910 

Deflected    perpendicular 
to  axis  of  pins. 

? 

434 


A  PPLIE  /)   ME  CHA  A  ICS. 


TABULATION   OF   EXPERIMENTS   ON   WR OUGHT-IRON  COLUMNS 
WITH    3^-INCH    PIN-ENDS. 


Length, 

Ultimate 
Strength. 

Centre 

Sec- 

No. of 
Test. 

Style  of  Column. 

to 
Centre 

tional 
Area. 

T/-\t<i  1 

Lbs. 

Manner  of  Failure. 

of  Pins. 

i  otai. 
Lbs. 

per 

In. 

Sq.  In. 

Sq.  In. 

Set-  G. 

1651 

320.00 

20.954 

540000 

25770 

Deflected      m     diagonal 
direction. 

1652 

l«^6                T 

320.10 

20.613 

535000 

25950 

Sheared    rivets    in    eye- 



'  2*  '  \ 

plates. 

746 

SetiH. 

159-20 

7.628 

258700 

339X0 

Deflected    perpendicular 
to  axis  of  pins. 

747 

?^i- 

f     Tt 

159-27 

8.056 

294700 

36580 

Do.             do. 

748 

^-8*44 

239.60 

7.621 

260000 

34120  |             Do.              do. 

749 

i 

239.60 

7.621 

254600  j     33410    Deflected      in     diagonal 

direction. 

1648 

ffi 

/  —  * 

x- 

31610  1  Deflected   narallel   to  ?xis 

319.90 

7-7 

243000 

of  pins. 

1649 

3^9-85 

7  '673 

229200 

29870 

Deflected      in     diagonal 
direction. 

740 

I 

i59-9o 

7-645 

262500 

34340 

Deflected     perpendicular 

Set  I. 

to  axis  of  pins. 

74  l 

(swelled.) 

i59-9o 

7.624 

255650 

33530 

Do.              do. 

739 

~~ 

i        |~~t 

239.70 

7-51? 

251000 

33390 

Deflected   parallel  to  axis 

1 

of  pins. 

75° 

(•        |  3  '—»$    "U 

239.70 

7  -S31 

259000 

34390 

Deflected     perpendicular 

. 

to  axis  oi  pins. 

1643 

_ 

1       _l 

319.80 

7.691 

237200 

30840 

Deflected  parallel  to  axis 

r»f    r»in<* 

1644 

1 

319.92 

7.702 

237000 

30770    Deflected      in      diagonal 
direction. 

< 

• 

1640 

J-K 

199.84 

"•944, 

403000 

33740 

Deflected     perpendicular 

~i      p* 

to  axis  of  pins. 

1641 
1634 

4-  - 

200.00 
300.00 

12.302 
12.148 

426500 
408000 

34670 
33630 

Deflected      in      diagonal 
direction. 
Deflected     perpendicular 

7 

to  axis  of  pins. 

1635 

_i 

3OO.OO 

12.175 

395000 

32440 

Do.              do. 

r 

WROUGHT-1RON   COLUMNS. 


435 


TABULATION   OF    EXPERIMENTS   ON   WROUGHT-IRON   COLUMNS 
WITH    3J-INCH    PIN-ENDS. 


Length, 
Centre 

Sec- 

Ultimate 
•    Strength. 

- 

No.  of 
Test. 

Style  of 

Column. 

to 
Centre 

tional 
Area. 

Total, 

Lbs. 

Manner  of  Failure. 

of  Pins. 

Lbs. 

per 

In. 

Sq.  In. 

Sq.  In. 

Sejt  M. 

(swelled.) 

1638 

1 

199-25 

12.366 

385000 

3«30 

Deflected    perpendicular 
to  axis  of  pins. 

1639 

n 

*"o 

199.50 

12.659 

405000 

31990 

Do.             do. 

1636 

I 

300.20 

11.920 

391400 

32830 

Deflected      in     diagonal 
direction. 

l637 

.  

J 

300.15 

11.932 

390700 

32740 

Do.             do. 

1630 

1 

*rf 

ik   s!  s_l 

300.00 

17.622 

461500 

26190 

Deflected    perpendicular 
to  axis  of  pins. 

itS 

» 

1631 

1 

300.00 

17.231 

485000 

28150 

Do.             do. 

1632 

|p£-10 

I 

.2 

4 

300.00 

17-57° 

306000 

17420 

Do.            do. 

^|      1 

*^o 

1633 

jr^-f 

300.00 

17.721 

307000 

17270 

Do.            do. 

K 

I1     " 

The  remainder  of  the  tests  of  this  series  of  seventy-four 
columns  is  reported  in  the  volume  for  1884. 

The  only  portion  of  the  description  that  it  is  worth  while 
to  quote  is  the  following,  as  the  tests  were  made  in  a  similar 
way  to  what  has  been  already  described  : 

"  Sixteen  posts  were  tested  with  flat  ends ;  eighteen  were 
tested  with  3^-inch  pin-ends. 


436 


APPLIED    MECHANICS. 


"  The  pins  were  placed  in  the  centre  of  gravity  of  cross- 
section,  except  two  posts  of  set  K,  which  had  the  pins  in  the 
centre  of  gravity  of  the  channel-bars,  giving  an  eccentric  bear- 
ing to  these  columns,  on  account  of  the  continuous  plate  on 
one  side  of  the  channel-bars." 


TABULATION   OF   EXPERIMENTS    ON   WROUGI IT-IRON   COLUMNS 
WITH   FLAT   ENDS. 


Ultimate 

Total 

Sec- 

Strength. 

No.  of 
Test. 

Style  of  Column. 

Length. 

tional 
Area. 

Total, 

Per 

Number  of  Failure. 

Sq.  In., 

Ft.    In. 

Sq.  In. 

Lbs. 

377 
378 

SetB. 

1 

*" 

10     7.90 
10     7.90 

12.08 
ii.  ii 

383200 
372900 

31722 
33564 

Buckling-plate     D    be- 
tween the  riveting. 
Buckling-plates. 

"4 

I 

379 

SetE. 

-84 

^ 

13  i  i.  80 
13  11.80 

17.01 
17.80 

633600 

34950 
35595 

Buckling  -  plates      be' 
tween  the  riveting. 
Triple  flexure. 

346 

13  11.9 

15.74 

517000 

32846 

Buckling-plates. 



v^"  "T" 

347 

£  1 

,f 

13  11.65 

15.84 

555200 

35050 

Do.             do. 

Set  F. 

.a//'         ^J 

7j6 

342 

<—  7^ 

20     7  .  63 

15.68 

517500 

33003 

Deflecting  upward. 

344 

w 

20     7  .  80 

15-56 

536900 

345°5 

Buckling-plates. 

348 

T3  "-75 

21  .02 

708000 

33682 

Buckling-plates. 

ll          ^           ^ 

1  

349 

;i«"    ?| 

1 

X3   "-75 

21  .46 

709500 

33061 

Triple  flexure. 

343 

SetG.  jfc-Ka"        § 
_>-    6.90% 

it> 

20     7.60 

20       7.63 

21.20 
21.49 

700000 
729450 

330'9 
33943 

Deflecting  upward. 
Deflecting  downward. 

^8" 

WRO  UC,H 7 '-IRON   COL  UMNS. 


437 


TABULATION   OF   EXPERIMENTS   ON   WROUGHT-IRON   COLUMNS 
WITH    FLAT   ENDS. 


Ultimate 

Total 

Sec- 

Strength. 

No.  of 
Test. 

Style  of  Column. 

Length. 

tional 
Area. 

Manner  of  Failure. 

Total, 

Per  Sq. 

Ft.    In. 

Sq.  In. 

Lbs. 

In.,  Ibs. 

339 

20    7-94 

12.64 

412900 

32666 

Deflecting  upward. 

"         7*    1  —  T 

***  H 

SetK. 

T 

340 

er= 

±£=LI 

20    7.94 

12.74 

431400 

33862 

Do.             do. 

Latticed 

%,'plate.71 

337 
338 

SetN. 

-"I! 

25     7-75 
25     7-88 

16.99 
17.40 

582400 
580000 

34279 

33333 

Deflecting     downward 
and  sideways. 
Deflecting      diagonally 
channel  B  and  lattic- 
ing on    the    concave 

Latticed 

side. 

TABULATION   OF   EXPERIMENTS   ON   WROUGHT-IRON    COLUMNS 
WITH    3^-INCH    PIN-ENDS. 


Length, 

Ultimate 

Centre      Sec- 

Strength. 

No.  of 
Test. 

Style  of  Column. 

to      i    tional 

Centre     Area, 
of  Pins. 

Manner  o(  Failure. 

Total,    Per  Sq. 

Ft.    In. 

Sq.   In. 

Lbs.     In.,  Ibs. 

l 

368 

1 

15      o.i 

11.42 

379200       33205 

Hor.  deflection  perpen- 
dic.  to  plane  of  pins. 

356 

Set  B. 

jji 

if 

20         0.0 

11.42 
ii  .42 

342000 

29947 

Do.              do. 

357 

20         0.0 

11.31 

330100 

29186 

Do.              do. 

% 

< 

371 

9     ir-9 

9.14 

286100 

31302 

Buckling  -  plates      be- 
tween rivets. 

372 

J 

IO         O.O 

10.07 

319200 

31698 

Do.              do. 

370 

kf 

i 

4 

a 

15       o.o 

9.21 

291500 

31650  I  Hor.  deflec.  and  buck- 

369 

Set  C. 

•» 

J  (. 
o. 

»4 

L_ 

15       o.o 

9-44 

290000 

30720 

ling  between  rivets. 
Do.              do. 

354 

'•* 

20       o.o 

9.24 

267500 

28950 

Triple  flexure. 

365 

20       o.o 

9-36 

279700 

29879 

Hor.  deflection. 

438 


APPLIED    MECHANICS. 


TABULATION   OF   EXPERIMENTS   ON  WROUGHT-IRON  COLUMNS 
WITH    3^-INCH   PIN-ENDS. 


Style  of  Column. 

Ultimate 

Length, 
Centre 

Sec- 

Strength. 

No.  of 
Test. 

to 
Centre 

tional 
Area. 

Total, 

Per 

Manner  of  Failure. 

of  Pins. 

Sq.  In., 

Ft.    In. 

Sq.  In. 

Lbs. 

I          l 

_l 

360 
361 

<-^t> 

SetD. 

l| 

<*£ 

*3     4-I3 
13     4.00 

15-34 
15.40 

475000 
485000 

30965 
3*494 

Deflecting    upward    in 
plane  of  pins. 
Hor.  deflection  perpen- 
dicular   to    plane    of 
pins. 

358 

Sot  E. 

, 

h 

20    o.o 

17.77 

570000 

32077 

Hor.  deflection  perpen- 
dicular   to    plane    of 

«." 

"jj 

1 

l 

pins. 

359 

-  s"\ 

4* 

20    o.o 

17.22 

5554oo 

32253 

Do.             do. 

350 

^ 

C 

20      0.25 

12.48 

202700 

16242 

Hor.    deflection,    con- 

C0| 

o- 

cave  on  lattice  side. 

351 

SetK. 

!l 

IJ 

00 

1 

20      0.00 

10.84 

208200 

10207 

Do.             do. 

352 

i 

"\ 

Lt 

5 

2O      O.25 

12.65 

350000 

27668 

Do.             do. 

353 

51 

E 

2O      O.25 

12.7 

390400 

30596 

Hor.  deflection  perpen- 
.  dicular    to   plane    of 

••     -I- 

pins,    convex  on  lat- 

tice side. 

Besides  the  above,  there  are  four  tests  of  lattice  columns 
reported  in  Exec.  Doc.  36,  49th  Congress,  1st  session,  but  as 
these  columns  were  rather  poorly  constructed  and  form  rather 
special  cases  they  will  not  be  quoted  here. 

In  determining  the  strength  of  a  bridge  column  made  of 
channel-bars  and  latticing,  these  results  of  tests  on  full-size 
columns  furnish  us  the  best  data  upon  which  to  base  our  con- 
clusions. 


WROUGHT-IRON   COLUMNS. 


439 


In  the  Trans.  Am.  Soc.  C.  E.  for  April,  1888,  Mr.  C.  L.  Stro- 
bel  gives  an  account  of  his  tests  on  wrought-iron  Z-bar  columns, 
from  which  the  following  is  condensed,  viz.:  The  Z-irons  used 
in  making  the  columns  were  2^X3X2^  inches  in  size,  and  & 
inch  thick. 

Two  columns  were  about  n  ft.  long,  two  15  ft.,  two  19  ft., 
three  22  ft.,  three  25  ft.,  and  three  28  ft.,  a  total  of  fifteen  columns. 
The  table  of  results  follows: 


Ultimate 

Ultimate 

Strength, 

Length, 

Sectional 
Area, 

Strength 
by  Tests 

/ 

Q 

Strobel's 

per  Sq.  In. 

Formula 

per  Sq.  In. 

Inches. 

Sq.  Ins. 

Lbs. 

Lbs. 

13'i 

9-435 

36800 

64 



131* 

34600 

64 

— 

180 

9.480 

34600 

88 

35000 

180 

9.280 

36600 

88 

35000 

228f 

9.241 

33800 

112 

32200 

228f 

10.  104 

33700 

112 

32200 

264 

9.286 

30700 

129 

29900 

264' 

9.286 

29500 

129 

29900 

264 

9.286 

30700 

129 

29900 

300 

9.156 

28100 

I46 

27750 

300 

9-456 

28000 

I46 

27750 

300 

9.516 

28400 

I46 

27750 

336 

9-375 

27700 

164 

25500 

336 

9-643 

28000 

l64 

25500 

336 

9-375 

27600 

164 

25500 

The  following  table  shows  the  results  of  compression  tests 
made  in  the  engineering  laboratories  of  the  Massachusetts  In- 
stitute of  Technology  upon  some  wrought-iron  pipe  columns. 
They  were  tested  with  the  ordinary  cast-iron  flange-coupling 
screwed  on  to  the  ends,  bearing  against  the  platforms  of  the 
testing- machine,  which  were  adjustable,  inasmuch  as  they  were 
provided  with  spherical  joints. 


440 


APPLIED    MECHANICS. 


The  tests  of  full-size  wrought-iron  columns  made  by  Mr.  G. 
Bouscaren,  are  given  in  the  Report  of  the  Progress  of  Work  on 
the  Cincinnati  Southern  Railway,  by  Thos.  D.  Lovett,  Nov.  i, 

1875- 

In  Heft  IV  (1890)  of  the  Mittheilungen  der  Materialprii- 
fungsanstalt  in  Zurich  is  given  an  account  of  a  large  number 
of  tests  of  wrought-iron  and  steel  columns  of  the  following 
forms,  viz.:  i°.  Angle-irons;  2°.  Tee  iron;  3°.  Channel- bars ; 
4*.  Two  angle-irons  riveted  together ;  5°.  Four  angle-irons 
riveted  together;  6°.  Two  channel-bars  riveted  together; 
7°.  Two  tee  irons  riveted  together  ;  also  quite  a  number  of  tests 
of  columns  of  some  of  these  forms  subjected  to  eccentric 
loads,  the  eccentricity  of  the  load  being,  in  some  cases,  as  much 
as  8  cm.  (3".  15).  The  columns  tested  were  of  a  variety  of 
lengths,  the  longest  ones  being  560  cm.  (18.37)  ^eet  l°ng- 

In  Heft  VIII  (1896)  of  the  same  Mittheilungen  is  an  ac- 
count of  another  set  of  tests  of  columns  of  the  above-described 
forms.  The  results  of  these  valuable  tests  will  not  be  quoted 
here,  but  for  them  the  reader  is  referred  to  the  Mittheilungen 
themselves. 


"o 

u 

V 

V 

I 

_H 

•2 

i 

•a 
a 

•5  c 

ominal  Size 
Pipe. 

iside  Diame 

utside  Diam 
eter. 

iameter  of 
Flanges. 

li 

£"0 

auge  Lengtl 

aximum  Lo; 

rea  of  Cross 
section. 

'aximum  Lo 
per  Sq.  In. 

ompression, 
Modulus  of 
Elasticity. 

o  o  2 

u   c'o 

* 

M 

0 

C 

O 

s 

< 

2 

U 

^  TQ. 

In. 

In. 

In. 

In. 

In. 

In. 

Lbs. 

Sq.  In. 

Lbs. 

2 

2.06 

2-37 

7 

I 

51 

30000 

i.  08 

27800 

24300000 

88.8 

2 

2.04 

2-39 

7 

69 

1 

51 

29800 

1  .22 

24500 

222OOOOO 

89.1 

25 

2.50 

2.89 

8 

93 

86 

34500 

1.65 

20900 

25200000 

98.1 

2i 

2.48 

2.88 

8 

93: 

86 

37000 

1.68 

22000 

259OOOOO 

98.4 

3 

3.06 

3-44 

8f 

93 

| 

86 

45500 

1.94 

23500 

27700000 

81.4 

3 

3.48 

8| 

93' 

86 

51000 

2.01 

25300 

25IOOOOO 

80.5 

3  '.60 

4.00 

105! 

100.5 

55000 

2-39 

23000 

25200000 

78.2 

32 

3-59 

3-99 

9i 

1:051 

100.5 

65000 

2-39 

27200 

246OOOOO 

78.5 

4 

4.07 

4'53 

9fr 

100.5 

80000 

3-" 

25700 

258OOOOO 

77-1 

4 

4.09 

4-50 

"7ft 

100.5 

69000 

2.76 

25000 

24900000 

77-3 

§  224.      Transverse     Strength    of   Wrought-iron. — 

Wrought-iron  owes  its  extensive  introduction  into  con- 
struction as  much  or  more  to  the  efforts  of  Sir  William 
Fairbairn  than  to  anyone  else;  and  while  he  was  furnishing 


TRANSVERSE   STRENGTH  OF   WROUGHT-IRON.         441 

the  means  to  Eaton  Hodgkinson  to  make  extensive  experiments 
on  cast-iron  columns,  and  while  he  made  experiments  himself 
on  cast  iron  beams,  which  were  in  use  at  that  time,  he  also 
carried  on  a  large  number  of  tests  on  beams  built  of  wrought- 
iron,  more  especially  those  of  tubular  form,  and  those  having 
an  I  or  a  T  section,  and  made  of  pieces  riveted  together.  In 
his  book  on  the  "  Application  of  Cast  and  Wrought  Iron  to 
Building  Purposes "  he  gives  an  account  of  a  large  number 
of  these  experiments,  including  those  made  for  the  purpose  of 
designing  the  Britannia  and  Conway  tubular  bridges,  a  fuller 
account  of  which  will  be  found  in  his  book  entitled  "  An  Ac- 
count of  the  Construction  of  the  Britannia  and  Conway  Tubular 
Bridges."  In  the  first-named  treatise  he  urges  very  strongly 
the  use  of  wrought-iron,  instead  of  cast-iron,  to  bear  a  trans- 
verse load. 

Fairbairn  tested  a  number  of  wrought-iron  built-up  beams, 
but  they  were  of  small  dimensions  and  are  hardly  comparable 
with  those  used  in  practice. 

In  the  light  of  the  tests  made  upon  wrought-iron  columns, 
it  is  evident  that  the  compressive  strength  of  wrought-iron  is 
less  than  the  tensile  strength.  Hence  we  should  naturally  ex- 
pect that  the  modulus  of  rupture  would  be,  in  all  cases,  greater 
than  the  compressive  strength,  and  that  it  might  or  might  not 
be  greater  than  the  tensile  strength  of  the  iron.  Of  course 
the  modulus  of  rupture  varies  very  much  with  the  shape  of  the 
cross-section,  for  the  same  reasons  as  were  explained  in  the 

paragraph  191,  i.e.,  that  the  formula  M  =  f—  assumes  Hooke's 

law,  "the  stress  is  proportional  to  the  strain,"  to  hold,  and  that 
this  is  not  true  near  the  breaking-point. 

The  value  of  the  modulus  of  rupture  is  also  influenced 
by  the  reduction  in  the  rolls,  and  hence  somewhat  by  the  size 
of  the  beam. 

Small  round  or  rectangular  bars  tested  for  transverse 
strength  show  a  modulus  of  rupture  very  much  in  excess  of 
the  compressive  strength  per  square  inch  of  the  iron,  and  ex- 
ceeding very  considerably  even  the  tensile  strength. 

While   a   great   many  tests   of  such   specimens   have    been 


442  APPLIED    MECHANICS. 

made,  none  will  be  quoted  here,  but  the  last  five  tests  of  the 
table  on  page  542  show  that  for  a  wrought-iron  having  a  ten- 
sile strength  per  square  inch  from  58700  to  60250  pounds,  mo- 
duli of  rupture  were  obtained  from  Soooo  to  90000  pounds, 
as,  the  number  of  turns  of  these  rotating  shafts  being  com- 
paratively small,  the  breaking-loads  were  not  far  below  the 
quiescent  breaking  loads.  On  the  other  hand  the  moduli  of 
rupture  of  I  beams  and  other  shapes  used  in  building  have 
very  much  lower  values,  but  for  these,  tests  will  be  cited. 

As  to  experiments  on  large  beams,  we  have : 

i°.  Some  tests  made  by  Mr.  William  Sooy  Smith  and  jy 
Col.  Laidley  at  the  Watertovvn  Arsenal. 

2°.  Some  tests  made  in  Holland  on  iron  and  steel  beams, 
an  account  of  which  is  given  in  the  Proceedings  of  the  Brit- 
ish Institute  of  Civil  Engineers  for  1886,  vol.  Ixxxiv.  p.  412  et 
seq. 

3°.  Some  tests  made  in  the  laboratory  of  Applied  Mechan- 
ics of  the  Massachusetts  Institute  of  Technology,  on  iron  and 
steel  I  beams. 

4°.  Tests  made  by  the  different  iron  companies  upon  beams 
of  their  own  manufacture,  and  recorded  in  their  respective 
hand-books. 

Mr.  Smith's  tests  are  recorded  in  Executive  Document  23, 
46th  Congress,  second  session. 

5°.  In  Heft  IV  (1890)  of  the  Mittheilungen  der  Material- 
priifungsanstalt  in  Zurich  will  be  found  accounts  of  tests 
made  by  Prof.  Tetmajer  upon  the  transverse  strength  of 
I  beams,  of  deck-beams,  and  of  plate  girders. 

The  results  of  these  tests  will  be  given  in  the  table  on  top 
of  page  443- 

Specimens  cut  from  the  flanges,  and  also  from  the  webs  of 
the  last  seven  of  these  beams,  were  tested  for  tension.  In  the 
case  of  those  cut  from  the  flanges,  the  tensile  strength  varied 


TRANSVERSE   STRENGTH  OF   WROUGHT-IRON.        443 


Depth. 
(Inches.) 

Moment  of 
Inertia. 
(Inches)*. 

Span. 
(Inches.) 

Modulus  of 
Rupture. 
^Lbs.  per  Sq.  In.) 

Modulus  of 
Elasticity. 
(Lbs.  per  Sq.  In.) 

7.87 
7.87 
3-93 
5-9i 
7.87 
9-45 
n.8i 

52.04 
52.04 

4.13 
17-85 
5T-95 
103.  .-.4 

62.96 
62.96 
3i-44 
47.28 
62.96 
75.60 
94.48 

51190 

56453 
62852 

56453 
53894 
51619 

CT  t8^? 

27501500 
28937700 
28767101 
28212500 
28226700 
2737350o 

4  34 

from  50200  in  the  1 5". 75  beam  to  57300  pounds  per  square  inch 
in  the  3^.93  beam.  On  the  other  hand,  in  the  case  of  the 
specimens  cut  from  the  web,  the  tensile  strengths  varied  from 
44900  in  the  I  i^.Si  beam  to  54400  pounds  per  square  inch  in  the 
3"-93  beam,  the  contraction  of  area  per  cent  varying  from  23.6 
to  32.1  per  cent  in  the  flanges,  and  from  12.5  to  15.9  per  cent 
in  the  web. 

The  results  obtained  with  the  deck-beams  are  as  follows : 


Depth. 
(Inches.) 

Moment  of 
Inertia. 
(Inches)4. 

Span. 
(Inches.) 

Modulus  of 
Rupture. 
(Lbs.  per  Sq.  In.) 

Modulus  of 
Elasticity. 
(Lbs.  per  Sq.  In.) 

4-93 

19.88 

70.86 

56170 

25112500 

4.26 

9-38 

59.06 

48920 

25823500 

3-52 

5-33 

47.24 

55320 

25596000 

3.48 

4.71 

39-37 

54180 

26804700 

2.36 

1.30 

31.50 

52760 

24202400 

'•93 

0.60 

23.62 

58160 

Tensile  tests  of  specimens  cut  from  these  deck-beams 
showed  tensile  strengths  of  from  47540  in  the  i".93  beam  to 
54750  pounds  per  square  inch  in  the  2". 36  beam,  and  contrac- 
tions of  area  of  from  14.1  per  cent  to  18.4  per  cent. 

The  results  obtained  with  the  plate  girders  are  as  follows, 
viz. : 


Depth  of 
Web. 
(Inches.) 

Modulus  of 
Rupture. 
(Lbs.  per  Sq. 
In.) 

Modulus  of 
Elasticity. 
(Lbs.  per  Sq. 
In.) 

Depth  of 
Web. 
(Inches.) 

Modulus  of 
Rupture. 
(Lbs.  per  Sq. 
In.) 

Modulus  of 
Elasticity. 
(Lbs.  per  Sq. 
In.) 

iS-75 
iS-75 

19.69 
19.69 

51480 
53180 
SH?6 
52610 

26449200 
25539100 
24813900 
25605500 

23.62 
23.62 
27.56 
27-56 

52760 
48490 
47780 
46500 

26321200 

26548700 
25667100 
26776300 

444 


APPLIED    MECHANICS. 


The  tensile  strength  of  the  material  of  the  webs  varied  from 
29860  to  41240  pounds  per  square  inch,  while  the  contraction  of 
area  was  only  0.4  per  cent.  The  tensile  strength  of  the  material 
of  the  flange-plates  was  51050  pounds  per  square  inch,  with  a 
contraction  of  area  of  17  percent.  The  tensile  strength  of  the 
angle-irons  was  46357  pounds  per  square  inch,  with  a  contraction 
of  area  of  14  per  cent. 

The  following  table  gives  the  results  that  have  been  obtained 
in  the  tests  that  have  been  made  upon  wrought-iron  I  beams  in 
the  laboratory  of  Applied  Mechanics  of  the  Massachusetts  Insti- 
tute of  Technology.  This  table  will  give  a  fair  idea  of  the  strength 
and  elasticity  of  such  beams. 

TESTS  OP  WROUGHT-IRON  BEAMS  MADE  IN  THE  LABORATORY  OF  APPLIED 
MECHANICS  OF  THE  MASSACHUSETTS  INSTITUTE  OF  TECHNOLOGY, 
ALL  LOADED  AT  THE  CENTER. 


No. 
of 
Test. 

Ins. 

Moment 
of 
Inertia. 

Span. 
Ft.    Ins. 

Break- 
ing Load. 

Lbs. 

Moduli 
of 
Rupture. 
Lbs.  per 
Sq.  In. 

Moduli  of 
Elasticity. 
Lbs.  per 
Sq.  In. 

Remarks. 

121 

6 

24.41 

12            0 

9500 

42386 

26679000 

From  Phoenix  Co. 

124 

7 

43-5° 

14         o 

14100 

48082 

28457000 

<  •           a 

126 

5 

12.47 

12            0 

6450 

46624 

29549000 

<  t                           t  1                         ! 

209 

7 

44-93 

13        8 

1  2  2OO 

39670 

31057000 

t   •                            1  I 

211 

8 

67.32 

13        8 

I7OOO 

42000 

28532000 

«                       <  (                      4 

215 

9 

no.  78 

14         8 

23000 

41680 

27165000 

'  .                       1  '                      ' 

227 

8 

61  .20 

13         6 

18300 

49640 

27397000 

From  Belgium. 

230 

9 

86.41 

13        8 

21300 

45850 

27365000 

<  *           <  > 

235 

9 

40.91 

13        8 

13800 

49140 

27923000 

«  E           <  • 

253 

7 

43-05 

14        8 

II3I9 

40660 

28045000 

From  Phoenix  Co. 

256 

8 

66.56 

14        8 

14547 

38460 

28187000 

i  •          « 

263 

9 

108.67 

14        8 

19694 

36160 

27050000 

«           « 

291 

7 

45-96 

14         6 

IO7OO 

36340 

26790000 

•  i           <• 

292 

8 

66.39 

14        6 

14300 

38200 

27380000 

«           «  • 

294 

9 

92.89 

14        6 

I92OO 

41470 

27050000 

•i          '-• 

338 

,  6 

25.92 

i4        7 

72OO 

37800 

27860000 

N.  T.  Steel  &  Iron  Co. 

34i 

7 

46.73 

12         II 

13600 

40600 

27410000 

I  •              It        H        «          « 

345 

8 

7I-25 

14        7 

15400 

38400 

26940000 

«i            ,:       T,       ,t        ,, 

379 

7 

48.84 

12         IO 

15500 

44300 

26170000 

STEEL.  445 


§  225.  Steel. — While  steel  is  a  malleable  compound  of  iron, 
with  less  than  2  per  cent  of  carbon  and  with  other  substances, 
the  definition  recommended  by  an  international  committee  of 
metallurgists  in  1876,  and  used  to  some  extent  in  German  and 
Scandanavian  countries,  is  different  from  that  in  general  use  in 
English-speaking  countries,  and  in  France. 

The  definition  recommended  by  the  international  committee 
may  be  found  in  the  Trans.  Am.  Inst.  Min.  Engrs.  for  October, 
1876,  and  is  in  the  following: 

i°.  That  all  malleable  compounds  of  iron  with  its  ordinary 
ingredients,  which  are  aggregated  from  pasty  masses,  or  from 
piles,  or  from  any  form  of  iron  not  in  a  fluid  state,  and  which 
will  not  sensibly  harden  and  temper,  and  which  generally 
resemble  what  is  called  "wrought-iron,"  shall  be  called  weld-iron 
(Schweisseisen). 

2°.  That  such  compounds,  when  they  will,  from  any  cause, 
harden  and  temper,  and  which  resemble  what  is  now  called 
puddled  steel,  shall  be  called  weld-steel  (Schweissstahl). 

3°.  That  all  compounds  of  iron,  with  its  ordinary  ingredients 
which  have  been  cast  from  a  fluid  state  into  malleable  masses, 
and  which  will  not  sensibly  harden  by  being  quenched  in  water 
while  at  a  red  heat,  shall  be  called  ingot-iron  (Flusseisen). 

4°.  That  all  such  compounds,  when  they  will,  from  any  cause, 
so  harden,  shall  be  called  ingot-steel  (Flussstahl). 

On  the  other  hand,  in  English-speaking  countries,  those 
compounds  which  have  been  aggregated  from  a  pasty  mass, 
usually  in  the  puddling -furnace,  and  which  contain  slag,  are 
generally  called  wrought-iron,  while  those  which  have  been  cast 
from  a  molten  state  into  a  malleable  mass  are  generally  called  steel. 

While  this  classification  is  not  perfect,  it  states  the  most 
common  practice  in  a  general  way.  Exceptions,  two  of  which 
are  that  it  does  not  include  the  cases  of  cementation  steel  and 
of  puddled  steel,  will  not  be  discussed  here. 

In  view  of  the  above,  it  will  be  plain  that  what  is  commonly 


44°  APPLIED    MECHANICS. 

called  mild  steel  in  America,  would  be  called  ingot-iron  under 
the  definition  of  the  international  committee.  Steel  is  usually 
made  by  one  of  three  processes,  viz. :  the  crucible  process,  the 
Bessemer  process,  or  the  open-hearth  process. 

While  other  processes,  as  the  cementation  process  and  others, 
are  sometimes  used,  the  three  enumerated  above  are  in  most 
common  use  at  the  present  time. 

Crucible  Steel. — This  is  very  commonly  made  by  re-melting 
blister-steel  in  crucibles;  the  blister-steel  being  made  by  the 
cementation  process,  in  which  bars  of  very  pure  wrought-iron, 
especially  low  in  phosphorus,  are  heated  in  contact  with  charcoal 
until  they  have  absorbed  the  necessary  amount  of  qarbon. 

A  cheaper  process,  and  one  much  used  at  the  present  day, 
is  to  melt  a  mixture  of  charcoal  and  crude  bar-;ron  in  a 
crucible. 

Crucible  steel,  which  is  always  high-carbon  steel,  is  used  for 
the  finest  cutlery,  tools,  etc.,  and  "wherever  a  very  pure  and 
homogeneous  quality  of  steel  is  required. 

Bessemer  Steel. — In  the  Bessemer  process,  a  blast  of  air  is 
blown  into  melted  cast-iron,  removing  the  greater  part  of  its 
carbon  and  burning  out  more  or  less  of  the  other  ingredients. 
The  process  is  conducted  in  a  converter,  which  is  usually  so 
arranged  that,  when  the  operation  is  complete,  it  can  be  rotated 
around  a  horizontal  axis  to  such  an  extent  that  the  tuyeres  are 
above  the  surface  of  the  molten  steel,  and  the  blast  is  shut  off. 

In  the  acid  Bessemer  process,  the  lining  of  the  converter  is 
made  of  some  silicious  substance,  the  burning  of  silicon  being 
relied  upon  to  develop  a  sufficiently  high  temperature  to  keep 
the  metal  fluid. 

In  the  basic  Bessemer  process,  the  lining  of  the  converter  is 
of  such  a  nature  as  to  resist  the  action  of  basic  slags.  It  is  usually 
made  of  dolomite,  or  of  some  kind  of  limestone.  Burned  lime  is 
added  to  the  charge  to  seize  the  silicon  and  phosphorus,  the  latter 
serving  to  develop  a  sufficiently  high  temperature. 


STEEL.  447 


In  the  latter  part  of  the  operation,  the  phosphorus  is  largely 
burned  out,  whereas  in  the  acid  process,  in  order  to  produce  a 
steel  that  is  low  in  phosphorus,  it  is  necessary  to  use  a  pig-iron 
that  is  low  in  phosphorus. 

Open-hearth  Steel. — In  the  open-hearth  process,  a  charge  of 
pig-iron  and  scrap  is  placed  on  the  bed  of  a  regenerative  furnace, 
and  exposed  to  the  action  of  the  flame,  and  is  thus  converted  into 
steel. 

In  the  acid  open-hearth  process,  the  lining  of  the  furnace  is 
of  a  silicious  nature,  and  is  covered  with  sand,  while  in  the  basic 
it  is  usually  of  dolomite,  or  of  some  kind  of  limestone. 

Bessemer  and  open-hearth  steel  contain  more  impurities 
than  crucible  steel,  but  they  are  very  much  cheaper,  and  are 
just  as  suitable  for  many  purposes.  It  is  only  in  consequence 
of  their  introduction  that  steel  can  be  extensively  used  on  the 
large  scale,  as  crucible  steel  would  be  too  expensive  for  many 
purposes. 

Steel,  unlike  wrought-iron,  is  fusible;  unlike  cast-iron,  it  can 
be  forged;  and,  with  the  exception  of  the  harder  grades,  it  can 
be  welded  by  heating  and  hammering,  the  welding  of  high -carbon 
steel  in  large  masses  being  a  very  uncertain  operation,  though 
small  masses  can  be  welded  by  taking  proper  care. 

The  special  characteristic,  however,  is,  that,  with  the  exception 
of  the  milder  grades,  when  raised  to  a  red  heat  and  suddenly 
cooled,  it  becomes  hard  and  brittle,  and  that,  by  subsequent 
heating  and  cooling,  the  hardness  may  be  reduced  to  any  desired 
degree.  The  first  process  is  called  hardening  and  the  second 
tempering. 

The  principal  element  in  the  steels  that  are  ordinarily  used  is 
carbon;  nevertheless,  both  Bessemer  and  open-hearth  steel  con- 
tain also  silicon,  manganese,  sulphur,  phosphorus,  etc.,  which 
have  more  or  less  effect  upon  the  resisting  properties  of  the  metal. 
Sulphur,  silicon,  and  phosphorus  usually  come  from  the  ore, 
the  fuel,  and  the  flux,  while  manganese,  which  is  added,  operates, 


44-8  APPLIED    MECHANICS. 

among  other  things,  to  render  the  steel  ductile  while  hot,  and 
therefore  workable,  and  to  absorb  oxygen  from  the  melted  mass. 

Sulphur  is  injurious  by  causing  brittleness  when  hot,  and 
phosphorus  by  causing  brittleness  when  cold.  Phosphorus  is 
the  most  harmful  ingredient  in  steel,  so  that  when  steel  is  to  be 
used  for  structural  purposes,  it  is  important  to  have  as  little 
phosphorus  as  possible,  and  any  excess  of  phosphorus  is  not  to 
be  tolerated. 

The  injury  done  to  steel  plates  by  punching  is  greater  than 
that  done  to  iron  plates:  this  injury  can,  however,  be  removed 
by  annealing.  Steel  requires  greater  care  in  working  it  than 
iron,  whether  in  punching,  flanging,  riveting,  or  other  methods 
of  working;  otherwise  it  may,  if  overheated,  burn,  or  receive 
other  injury  from  careless  workmanship. 

The  chemical  composition  of  steel  is  one  important  element 
in  its  resisting  properties;  but,  on  the  other  hand,  the  mode  of 
working  also  has  a  great  influence  on  the  quality. 

The  introduction  of  the  Bessemer  process  was  quickly  fol- 
lowed by  the  general  use  of  steel  rails,  and  later,  as  this  and  the 
other  processes  for  making  steel  for  structural  purposes  have 
been  developed,  there  has  been  a  constant  increase  in  the  pur- 
poses for  which  steel  has  been  used. 

One  of  the  earlier  applications  was  to  the  construction 
of  steam-boilers,  steel  boiler-plate  displacing  almost  entirely 
wrought-iron  boiler-plate.  Of  late  years  the  development  of 
the  steel  manufacture  has  so  perfected,  and  at  the  same  time 
cheapened,  structural  steel  that  it  is  now  used  in  most  cases 
where  wrought-iron  was  formerly  employed.  Thus  the  eye-bars 
and  the  struts  of  bridges  are  almost  exclusively  made  of  steel, 
also  such  shapes  as  angle-irons,  channel-bars,  Z  bars,  tee  iron, 
I  beams,  etc.,  are  almost  exclusively  made  of  steel,  and  while 
steel  has  long  been  used  for  many  parts  of  machinery,  never- 
theless it  is  now  generally  used  in  many  cases  where  a  con- 
siderable fear  of  it  formerly  existed,  as  in  main  rods,  parallel 


^  TEEL.  449 


rods,  and  crank-pins,  and  in  a  large  number  of  parts  of  machinery 
subjected  to  more  or  less  vibration.  On  the  other  hand,  the 
steel  used  for  tools  is,  of  course,  high-carbon  steel. 

Tools  are  almost  always  made  of  crucible  steel,  and  they 
have  of  course  a  high  percentage  of  carbon,  a  high  tensile  strength, 
and  especially  should  they  be  capable  of  being  well  hardened 
and  taking  a  good  temper. 

The  usual  steel  of  commerce  may  be  called  carbon  steel, 
because,  although  it  always  contains  small  percentages  of  other 
ingredients,  nevertheless  carbon  is  the  ingredient  that  princi- 
pally determines  its  properties.  When  iron  or  steel  is  alloyed 
with  large  percentages  of  certain  substances,  the  resulting  al- 
loys enjoy  certain  special  properties,  and  these  alloys  still  bear 
the  name  of  steel.  Two  of  the  most  prominent  of  these  are 
manganese  steel  and  nickel  steel. 

Regarding  the  first  it  may  be  said  that  although  carbon  steel 
becomes  practically  useless  when  the  manganese  reaches  about 
ij  per  cent,  nevertheless  with  manganese  exceeding  about  7  per 
cent  we  obtain  manganese  steel  which  is  so  hard  that  it  is  exceed- 
ingly difficult  to  machine  it. 

The  alloy  that  has  come  into  most  prominent  notice  recently 
is  nickel  steel,  which  consists  most  commonly  of  a  carbon  steel 
with  from  0.2  to  0.4  per  cent  of  carbon  and  with  from  3  to  5 
per  cent  of  nickel.  With  this  amount  of  nickel  the  tensile  strength 
is  very  much  increased,  but  more  especially  is  the  limit  of  elasticity 
increased  by  a  very  large  amount;  and  while  the  contraction  of 
area  at  fracture  and  the  ultimate  elongation  per  cent  are  a  little 
less  than  that  of  carbon  steel  with  the  same  percentage  of  carbon, 
they  are  not  less  than  those  of  carbon  steel  of  the  same  tensile 
strength. 

It  is  used  for  armor-plates,  for  which  it  is  specially  suitable 
on  account  of  the  fact  that  the  nickel  renders  the  steel  more 
sensitive  to  hardening.  It  is  finding,  also,  a  great  many  other 
uses  to  which  it  is  specially  adapted  by  its  peculiar  properties. 


450  APPLIED   MECHANICS. 

It  has  been  used  for  bicycle-spokes,  for  shafts  for  ocean  steam- 
ships, for  piston-rods,  and  for  various  other  purposes.  Among 
the  many  examples  given  by  Mr.  D.  H.  Browne  in  a  paper  before 
the  American  Institute  of  Mining  Engineers  is  a  case  where  the 
presence  of  3.5  per  cent  nickel  increased  the  ultimate  strength 
of  0.2  per  cent  carbon  steel  from  55000  to  85000,  and  the  elastic 
limit  from  28000  to  48000  pounds  per  square  inch,  while  the 
contraction  of  area  at  fracture  was  only  decreased  from  60  per 
cent  to  55  per  cent. 

The  quality  of  steel  to  be  used  for  different  purposes  differs, 
and  while  the  specifications  for  any  one  purpose,  made  by  different 
engineers,  and  by  different  engineering  societies,  often  differ,  the 
work  of  the  American  Society  for  Testing  Materials  is  tending 
to  harmonize  them  as  far  as  possible.  The  result  of  their  efforts 
is  shown  in  the  following  set  of  specifications. 

AMERICAN    SOCIETY    FOR  TESTING  MATERIALS. 
SPECIFICATIONS  FOR  STEEL. 

STEEL  CASTINGS. 

Adopted  1901.     Modified  1905. 

PROCESS  OF  MANUFACTURE. 

1.  Steel  for  castings  may  be  made  by  the  open-hearth,  crucible, 
or    Bessemer    process.     Castings    to    be    annealed    unless    otherwise 
specified. 

CHEMICAL  PROPERTIES. 

2.  Ordinary  castings,  those  in  which  no  physical  requirements  are 
Ordinary      specified,  shall  not  contain  over  0.40  per  cent  of  carbon, 

Castings.        nor  ove].  Q  Qg  per  cent  Qf  phospnorus> 

3.  Castings  which  are  subjected  to  physical  test  shall  not  contain 
Tested         oyer  0.05  per  cent  of  phosphorus,  nor  over  0.05  per  cent 

Castings.        of  sulphun 

PHYSICAL  PROPERTIES. 

4.  Tested  castings  shall  be  of  three  classes:    "hard,"  " medium," 
Tensile         &nd  "soft."     The  minimum  physical  qualities  required  in 
Tests*  each  class  shall  be  as  follows : 


STEEL    CASTINGS. 


451 


Hard 

Medium 

Soft 

Castings. 

Castings. 

Castings. 

Tensile  strength,  pounds  per  square  inch  . 

85000 

70000 

60000 

Yield-point,  pounds  per  square  inch     . 

38250 

3I500 

27000 

Elongation,  per  cent  in  2  inches      .... 

15 

18 

22 

Contraction  of  area,  per  cent  

20 

25 

3° 

5.  A  test  to  destruction  may  be  substituted  for  the  tensile  test    in 
the  case  of  small  or  unimportant  castings    by  selecting 

three  castings  from  a  lot.     This  test  shall  show  the  material 
to  be  ductile  and  free  from  injurious  defects  and  suitable  for  the  pur- 
poses intended.     A  lot  shall  consist  of  all  castings  from  the  same  melt 
or  blow,  annealed  in  the  same  furnace  charge. 

6.  Large  castings  are  to  be  suspended  and  hammered  all  over. 
No  cracks,-  flaws,  defects,  nor  weakness  shall  appear  after    Percussive 
such  treatment.  Test- 

7.  A  specimen  one  inch  by  one-half  inch  (i"Xi")  shall  bend  cold 
around  a  diameter  of  one  inch  (i")  without  fracture  on    Bending 
outside  of  bent  portion,  through  an  angle  of  120°  for  "soft"    Test' 
castings  and  of  90°  for  "medium  "  castings. 

TEST  PIECES  AND  METHODS  OF  TESTING. 

8.  The  standard  turned  test  specimen   one-half  inch  (J")  diameter 
and  two  inch  (2")  gauged  length  shall  be  used  to  determine    Test  Speci- 
the  physical  properties  specified  in  paragraph  No.  4.     It    Tensiie"Test. 
is  shown  in  Fig.  i.     (See  page  398.) 

9.  The  number  of  standard  test  specimens  shall  depend  upon  the 
character  and  importance  of  the  castings.     A  test  piece 

shall  be  cut  cold  from  a  coupon  to  be  moulded  and  cast 
on  some  portion  of  one  or  more  castings  from  each  melt    JSens?  Spec" 
or  blow  or  from  the  sink-heads  (in  case  heads  of  sufficient 
size  are  used).     The  coupon  or  sink-head  must  receive  the  same  treat- 
ment as  the  casting  or  castings   before  the   specimen  is  cut  out,  and 
before  the  coupon  or  sink-head  is  removed  from  the  casting. 

10.  One  specimen  for  bending  test  one  inch  by  one-half  inch  (i"X  i") 
shall  be  cut  cold  from  the  coupon  or  sink-head  of  the  cast-  jest  specimen 
ing  or  dastings  as  specified   in  paragraph  No.   9.    The  for  Bending- 
bending  test  may  be  made  by  pressure  or  by  blows. 


452 


APPLIED    MECHANICS. 


n.  The  yield-point   specified  in  paragraph  No.  4  shall  be  deter- 
mined by  the  careful  observation  of  the  drop  of  the  beam 

Yield-point.  . J  . 

or  halt  in  the  gauge  of  the  testing-machine. 

12.  Turnings  from  tensile  specimen,  drillings  from  the  bending 

specimen,  or  drillings  from  the  small  test  ingot,  if  preferred 
chemical         by  the  inspector,  shall  be  used  to  determine  whether  or 

not  the  steel  is  within  the  limits  in  phosphorus  and  sulphur 
specified  in  paragraphs  Nos.  2  and  3. 

FINISH. 

13.  Castings  shall  be  true  to  pattern,  free  from  blemishes,  flaws, 
or  shrinkage  cracks.     Bearing-surfaces  shall  be  solid,  and  no  porosity 
shall  be  allowed  in  positions  where  the  resistance  and  value  of  the 
casting  for  the  purpose  intended  will  be  seriously  affected  thereby. 

INSPECTION. 

14.  The    inspector,  representing    the    purchaser,  shall    have    all 
reasonable  facilities  afforded  to  him  by  the  manufacturer  to  satisfy 
him  that  the  finished  material  is  furnished  in  accordance  with  these 
specifications.     All  tests  and  inspections  shall  be  made  at  the  place  of 
manufacture,  prior  to  shipment. 

STEEL  FORCINGS. 

Adopted  1901.     Modified  1905. 

PROCESS  OF  MANUFACTURE. 

1.  Steel  for  forgings  may  be  made  by  the  open-hearth,  crucible,  or 
Bessemer  process. 

CHEMICAL  PROPERTIES. 

2.  There  shall  be  four  classes  of  steel  forgings  which  shall  conform 
to  the  following  limits  in  chemical  composition: 


Forgings 
of  Soft  or 
Low-car- 
bon Steel. 

Forgings 
of  Carbon 
Steel  not 
Annealed. 

Forgings  of 
Carbon  Steel 
Oil-tempered 
or  Annealed. 

Loco- 
motive 
Forg- 
ings. 

Forgings  of 
Nickel  Steel, 
Oil-tempered 
or  Annealed. 

Phosphorus  shall  not  exceed 
Sulphur 
Manganese       "      "         " 
Nickel            

Per  Cent. 

0.  10 
0.  10 

Per  Cent. 
0.06 
0.06 

Per  Cent. 

0.04 
0.04 

Per  Ct, 

0.05 
0.05 
0.60 

Per  Cent. 
0.04 
0.04 

3  .  o  to  4  .  o 

Tensile 
Tests. 


PHYSICAL  PROPERTIES. 

3.  The    minimum   physical   qualities   required    of   the 
different-sized  forgings  of  each  class  shall  be  as  follows: 


STEEL  FORCINGS. 


453 


Tensile 
Strength. 
Lbs.  per 
Sq.  In. 

Yield- 
point. 
Lbs.  per 
Sq.  In. 

Elonga- 
tion in  2 
Inches. 
Per  Cent. 

Contrac- 
tion of 
Area. 
Per  Cent. 

SOFT  STEEL  OR  LOW-CARBON  STEEL. 

58000 

29000 

28 

35 

For  solid  or  hollow  f  orgings,  no  diameter 

- 

or  thickness  of  section  to  exceed  10". 

CARBON  STEEL  NOT  ANNEALED. 

75000 

375°° 

18 

30 

For  solid  or  hollow  forgings,  no  diameter 

or  thickness  of  section  to  exceed  10". 

Elastic 

Limit. 

CARBON  STEEL  ANNEALED. 

80000 

40000 

22 

35 

For  solid  or  hollow  forgings,  no  diameter 

or  thickness  of  section  to  exceed  10". 

75000 

375°° 

23 

35 

For  solid  forgings,  no  diameter  to  exceed 

20"  or  thickness  of  section  15". 

7OOOO 

35000 

24 

3° 

For  solid  forgings,  over  20"  diameter. 

CARBON  STEEL  OIL-TEMPERED. 

90000 

55000 

2O 

45 

For  solid  or  hollow  forgings,  no  diameter 

or  thickness  of  section  to  exceed  3". 

85000 

50000 

22 

45 

For  solid  forgings  of  rectangular  sections 

not  exceeding  6"  in  thickness  or  hol- 

low forgings,  the  walls  of  which  do  not 

exceed  6"  in  thickness. 

80000 

45000 

23 

40 

For  solid  forgings  of  rectangular  sections 

not  exceeding  10"  in  thickness  or  hol- 

low forgings,  the  walls  of  which  do  not 
exceed  10"  in  thickness. 

80000 

40000 

20 

25 

LOCOMOTIVE  FORCINGS. 

NICKEL  STEEL  ANNEALED. 

80000 

50000 

25 

45 

For  solid  or  hollow  forgings,  no  diameter 

or  thickness  of  section  to  exceed  10". 

8OOOO 

45000 

25 

45 

For  solid  forgings,  no  diameter  to  exceed 
20"  or  thickness  of  section  15". 

80000 

45000 

24 

40 

For  solid  forgings,  over  20"  diameter. 

NICKEL  STEEL,  OIL-TEMPERED. 

95000 

65000 

21 

50 

For  solid  or  hollow  forgings,  no  diameter 

or  thickness  of  section  to  exceed  3". 

9OOOO 

60000 

22 

50 

For  solid  forgings  of  rectangular  sections 

not  exceeding  6"  in  thickness  or  hol- 

low forgings,  the  walls  of  which  do  not 

exceed  6"  in  thickness. 

85000 

55ooo 

24 

45 

For  solid  forgings  of  rectangular  sections 

not  exceeding  10"  in  thickness  or  hol- 

low forgings,  the  walls  of  which  do  not 

exceed  10"  in  thickness. 

454  APPLIED    MECHANICS. 

4.  A   specimen   one    inch  by  one-half  inch    (i"Xj")    shall    bend 

cold  180°  without  fracture  on  outside  of  the  bent  portion, 
Test.  as  follows: 

Around  a  diameter  of  J  n ',  for  forgings  of  soft  steel. 

Around  a  diameter  of  ij",  for  forgings  of  carbon  steel  not  annealed. 

Around  a  diameter  of  ij",  for  forgings  of  carbon  steel  annealed,  if 
20"  in  diameter  or  over. 

Around  a  diameter  of  i",  for  forgings  of  carbon  steel  annealed, 
if  under  20"  diameter. 

Around  a  diameter  of  i",  for  forgings  of  carbon  steel,  oil  tempered. 

Around  a  diameter  of  J",  for  forgings  of  nickel  steel  annealed. 

Around  a  diameter  of  i",  for  forgings  of  nickel  steel,  oil  tempered, 

For  locomotive  forgjngs  no  bending  tests  will  be  required. 

TEST  PIECES  AND  METHODS  OF  TESTING. 

5.  The  standard  turned  test  specimen,  one-half  inch  (J")  diameter 
Test  sped-       and  two  (2")  gauged  length,  shall  be  used  to  determine 
site  Test.         the  physical  properties  specified  in  paragraph  No.  3. 

It  is  shown  in  Fig.  i.     (See  page  398.) 

6.  The  number  and  location  of  test  specimens  to  be  taken  from 

a  melt,  blow,  or  a  forging,  shall  depend  upon  its  character 
LoS«x>naofd  and  importance,  and  must  therefore  be  regulated  by 
hSensf  pec"  individual  cases.  The  test  specimens  shall  be  cut  cold 

from  the  forging  or  full-sized  prolongation  of  same  parallel 
to  the  axis  of  the  forging  and  half-way  between  the  centre  and  outside, 
the  specimens  to  be  longitudinal;  i.e.,  the  length  of  the  specimen  to 
correspond  with  the  direction  in  which  the  metal  is  most  drawn  out 
or  worked.  When  forgings  have  large  ends  or  collars,  the  test  specimens 
shall  be  taken  from  a  prolongation  of  the  same  diameter  or  section  as 
that  of  the  forging  back  of  the  large  end  or  collar.  In  the  case  of 
hollow  shafting,  either  forged  or  bored,  the  specimen  shall  be  taken  within 
the  finished  section  prolonged,  half-way  between  the  inner  and  outer 
surface  of  the  wall  of  the  forging. 

7.  The    specimen   for   bending   test   one    inch    by   one-half   inch. 
Test  Specimen  (I//Xj//)   shall  be  cut  as  specified  in  paragraph  No.   6. 
for  Bending.    The  bending  test  may  be  made  by  pressure  or  by  blows. 


OPEN-HEARTH  BOILER  PLATE  AND  RIVET  STEEL.     455 

8.  The  yield-point  specified  in  paragraph  No.  3  shall  be  determined 
by  the  careful  observation  of  the  drop  of  the  beam,  or 

halt  in  the  gauge  of  the  testing  machine.  Yield-point 

9.  The  elastic  limit  specified  in  paragraph  No.  3  shall  be  determined 
by  means  of  an  extensometer,  which  is  to  be  attached  to    Elastic 

the  test  specimen  in  such  manner  as  to  show  the  change    Limit. 

in  rate  of  extension  under  uniform  rate  of  loading,  and  will  be  taken 

at  that  point  where  the  proportionality  changes. 

10.  Turnings  from  the  tensile  specimen  or  drillings  from  the  bend- 
ing specimen  or  drillings  from  the  small  test  ingot,  if  pre- 
ferred by  the  inspector,  shall  be  used  to  determine  whether 

or  not  the  steel  is  within  the  limits  in  chemical  composition      na  ysis* 
specified  in  paragraph  No.  2. 

FINISH. 

11.  Forgings  shall  be  free  from  cracks,   flaws,   seams,   or  other 
injurious  imperfections,  and  shall  conform  to  the  dimensions  shown 
on  drawings  furnished  by  the  purchaser,  and  be  made  and  finished  in 
a  workmanlike  manner. 

INSPECTION. 

12.  The  inspector,  representing  the  purchaser,  sha.ll  have  all  reason- 
able facilities  afforded  him  by  the  manufacturer  to  satisfy  him  that 
the  finished  material  is  furnished  in  accordance  with  these  specifications. 
All  tests  and  inspections  shall  be  made  at  the  place  of  manufacture, 
prior  to  shipment. 

OPEN-HEARTH  BOILER  PLATE  AND  RIVET  STEEL. 

Adopted  1901. 

PROCESS  OF  MANUFACTURE. 

1.  Steel  shall  be  made  by  the  open-hearth  process. 

CHEMICAL  PROPERTIES. 

2.  There  shall  be  three  classes  of  open-hearth  boiler  plate  and 
rivet  steel;   namely,  flange,  or  boiler  steel,  fire-box  steel,  and  extra- 
soft  steel,  which  shall  conform  to  the  following  limits  in  chemical 
composition: 


APPLIED    MECHANICS. 


Flange  or 
Boiler  Steel. 
Per  Cent. 

Fire-box 
Steel. 
Per  Cent. 

Extra  Soft 
Steel. 
Per  Cent. 

Phosphorus  shall  not  exceed  . 

Sulphur           "      "        "      .      . 
Manganese 

("Acid     0.06 
\  Basic   o  .  04 
0.05 
o  30  to  o  60 

Acid     o  .  04 
Basic  0.03 
0.04 
o  30  to  o  co 

Acid     o  .  04 
Basic  0.04 
0.04 

Boiler-rivet 
Steel. 


3.  Steel   for   boiler    rivets    shall   be    of  the   extra-soft 
class  as  specified  in  paragraphs  Nos.  2  and  4. 


PHYSICAL  PROPERTIES. 
4.  The  three  classes  of  open-hearth  boiler  plate  and  rivet  steel- 


Tensile 

Tests. 

ities : 


namely,  flange  or  boiler  steel,  fire-box   steel,  and  extra- 
soft   steel — shall  conform  to  the  following  physical  qual- 


Flange  or 
Boiler  Steel. 

Fire-box 
Steel. 

Extra  Soft 
Steel. 

Tensile   strength,   pounds 
per  square  inch   . 
Yield-point,  in  pounds  per 
square  inch,  shall  not  be 
less  than  
Elongation,  per  cent  in  8 
inches  shall  not  be  less 
than 

55000  to  65000 

JT.S. 

2C 

52000  to  62000 

IT.  s. 

26 

45000  to  55000 
JT.S. 

28 

5.  For  material  less  than  five-sixteenths  inch  (&")  and  more  than 

three-fourths  inch  ( j")  in  thickness  the  following  modifica- 

in°Eio'n*ation  t*ons  s^a^  ^e  ma(^e  m  t^le  requirements  for  elongation: 
for  Thin  and          (#)  For  each  increase  of  one-eighth  inch  (£")  in  thick- 
ness  above  three-fourths   inch    (f")   a  deduction  of  one 
per  cent  (i%)  shall  be  made  from  the  specified  elongation. 

(b).  For  each  decrease  of  one-sixteenth  inch  (&")  in  thickness 
below  five-sixteenths  inch  (&")  a  deduction  of  two  and  one-half  per 
cent  (2^%)  shall  be  made  from  the  specified  elongation. 

6.  The  three  classes  of  open-hearth  boiler  plate  and  rivet  steel 
B  nd'n  shall  conform  to  the  following  bending  tests;   and  for  this 
Tests.             purpose  the  test  specimen  shall  be  one  and  one-half  inches 
(ii")  wide,  if  possible,  and  for  all  material  three-fourths  inch  (}")  or 
less  in  thickness  the  test  specimen  shall  be  of  the  same  thickness  as  that 


OPEN-HEARTH  BOILER  PLATE  AND   RIVET  STEEL. 

of  the  finished  material  from  which  it  is  cut,  but  for  material  more  than 
three-fourths  inch  }")  thick  the  bending-test  specimen  may  be  one- 
half  inch  (J")  thick: 

Rivet  rounds  shall  be  tested  of  full  size  as  rolled. 

(c).  Test  specimens  cut  from  the  rolled  material,  as  specified  above, 
shall  be  subjected  to  a  cold  bending  test,  and  also  to  a  quenched  bending 
test.  The  cold  bending  test  shall  be  made  on  the  material  in  the  con- 
dition in  which  it  is  to  be  used,  and  prior  to  the  quenched  bending  test 
the  specimen  shall  be  heated  to  a  light  cherry-red,  as  seen  in  the  dark, 
and  quenched  in  water  the  temperature  of  which  is  between  80°  and 
90°  Fahrenheit. 

(d).  Flange  or  boiler  steel,  fire-box  steel,  and  rivet  steel,  both  before 
and  after  quenching,  shall  bend  cold  one  hundred  and  eighty  degrees 
(180°)  flat  on  itself  without  fracture  on  the  outside  of  the  bent  portion. 

7.  For  fire-box  steel  a  sample  taken  from  a  broken  tensile-test 
specimen  shall  not  show  any  single  seam  or  cavity  more   Homogeneity 
than  one-fourth  inch  (J")  long  in  either  of  the  three  fractures  Tests- 
obtained  on  the  test  for  homogeneity  as  described  below  in  paragraph  12. 

TEST  PIECES  AND  METHODS  OF  TESTING. 

8.  The  standard  specimen  of  eight  inch  (8")  gauged  length  shall  be 
used   to    determine    the    physical   properties    specified    in 
paragraphs  Nos.   4  and   <.     The  standard  shape  of  the  men  for 

.      ,  ,     ,,  ,  ,  •      T^-        Tensile  Test. 

test  specimen  for  sheared  plates  shall  be  as  shown  in  rig. 
2.     (See  page  398.) 

For  other  material  the  test  specimen  may  be  the  same  as  for 
sheared  plates,  or  it  may  be  planed  or  turned  parallel  throughout  its 
entire  length ;  and  in  all  cases,  where  possible,  two  opposite  sides  of  the 
test  specimens  shall  be  the  rolled  surfaces.  Rivet  rounds  and  small 
rolled  bars  shall  be  tested  of  full  size  as  rolled. 

9.  One  tensile-test  specimen  will  be  furnished  from  each  plate  as 
it  is  rolled,  and  two  tensile-test  specimens  will  be  furnished 

from  each  melt  of  rivet  rounds.     In  case  any  one  of  these 
develops  flaws  or  breaks  outside  of  the  middle  third  of  its 
gauged  length,  it  may  be  discarded  and  another  test  specimen  sub- 
stituted therefor. 


APPLIED    MECHANICS. 


10.  For  material  three-fourths  inch  (-£")  or  less  in  thickness  the 

bending-test  specimen  shall  have  the  natural  rolled  surface 
mensS?orCI~  on  two  opposite  sides.  The  bending-test  specimens  cut 

from  plates  shall  be  one  and  one-half  inches  (ij")  wide, 
and  for  material  more  than  three-fourths  inch  (£")  thick  the  bending- 
test  specimens  may  be  one-half  inch  (J")  thick.  The  sheared  edges 
of  bending-test  specimens  may  be  milled  or  planed.  The  bending- 
test  specimens  for  rivet  rounds  shall  be  of  full  size  as  rolled.  The 
bending  test  may  be  made  by  pressure  or  by  blows. 

11.  One  cold-bending  specimen  and  one  quenched-bending  specimen 

will  be  furnished  from  each  plate  as  it  is  rolled.  Two 
Bending0  cold-bending  specimens  and  two  quenched-bending  speci- 

mens will  be  furnished  from  each  melt  of  rivet  rounds. 
The  homogeneity  test  for  fire-box  steel  shall  be  made  on  one  of  the 
broken  tensile-test  specimens. 

12.  The  homogeneity  test  for  fire-box  steel  is  made  as  follows:    A 

portion  of  the  broken  tensile-test  specimen  is  either  nicked 
Homogeneity  with  a  chisel  or  grooved  on  a  machine,  transversely  about 
Phi-box  a  sixteenth  of  an  inch  (;&")  deep,  in  three  places  about 

two  inches  (2")  apart.  The  first  groove  should  be  made 
on  one  side,  two  inches  (2")  from  the  square  end  of  the  specimen; 
the  second,  two  inches  (2")  from  it  on  the  opposite  side;  and  the  third, 
two  inches  (2")  from  the  last,  and  on  the  opposite  side  from  it.  The 
test  specimen  is  then  put  in  a  vise,  with  the  first  groove  about  a  quarter 
of  an  inch  (J")  above  the  jaws,  care  being  taken  to  hold  it  firmly.  The 
projecting  end  of  the  test  specimen  is  then  broken  off  by  means  of  a 
hammer,  a  number  of  light  blows  being  used,  and  the  bending  being 
away  from  the  groove.  The  specimen  is  broken  at  the  other  two 
grooves  in  the  same  way.  The  object  of  this  treatment  is  to  open 
and  render  visible  to  the  eye  any  seams  due  to  failure  to  weld  up,  or  to 
foreign  interposed  matter,  or  cavities  due  to  gas  bubbles  in  the  ingot. 
After  rupture,  one  side  of  each  fracture  is  examined,  a  pocket  lens 
being  used,  if  necessary,  and  the  length  of  the  seams  and  cavities  is 
determined. 

13.  For  the  purposes  of  this  specification  the  yield  -point  shall  be 

determined  by  the  careful  observation  of  the  'drop  of  the 
Yield-point.  ^eam  or  najt  m  tne  gauge  of  the  testing  machine. 


OPEN-HEARTH  BOILER   PLATE  AND   RIVET  STEEL.     459 

14.  In  order  to  determine  if  the  material  conforms  to  the  chemical 
limitations    prescribed    in    paragraph    2    herein,    analysis 

shall  be  made  of  drillings  taken  from  a  small  test  ingot.    aiemicai°r 
An  additional  check  analysis  may  be  made  from  a  tensile    Anal*sis- 
specimen  of  each  melt  used  on  an  order,  other  than  in  locomotive 
fire-box  steel.     In  the  case  of  locomotive  fire-box  steel  a  check  analysis 
may  be  made  from  the  tensile  specimen  from  each  plate  as  rolled. 

VARIATION  IN  WEIGHT. 

15.  The  variation  in  cross  section  or  weight  of  more  than  2j  per 
cent  from  that  specified  will  be  sufficient  cause  for  rejection,  except  in 
the  case  of  sheared  plates,  which  will  be  covered  by  the  following  per- 
missible variations: 

(e)  Plates  12  J  pounds  per  square  foot  for  heavier,  up  to  100  inches 
wide  when  ordered  to  weight,  shall  not  average  more  than  2^  per  cent 
variation  above  or  2\  per  cent  below  the  theoretical  weight.  When 
100  inches  wide  and  over,  5  per  cent  above  or  5  per  cent  below  the 
theoretical  weight. 

(/)  Plates  under  12 J  pounds  per  square  foot,  when  ordered  to 
weight,  shall  not  average  a  greater  variation  than  the  following: 

Up  to  75  inches  wide,  2  J  per  cent  above  or  2  J  per  cent  below  the  theo- 
retical weight.  Seventy -five  inches  wide  up  to  100  inches  wide,  5  per  cent 
above  or  3  per  cent  below  the  theoretical  weight.  When  100  inches 
wide  and  over,  10  per  cent  above  or  3  per  cent  below  the  theoretical 
weight. 

(g)  For  all  plates  ordered  to  gauge  there  will  be  permitted  an  average 
excess  of  weight  over  that  corresponding  to  the  dimensions  on  the  order 
equal. in  amount  to  that  specified  in  the  following  table: 

TABLE  OF  ALLOWANCES  FOR  OVERWEIGHT  FOR  RECTANGULAR  PLATES 
WHEN  ORDERED  TO  GAUGE. 

Plates  will  be  considered  up  to  gauge  if  measuring  not  over  -^ 
inch  less  than  the  ordered  gauge. 

The  weight  of  one  cubic  inch  of  rolled  steel  is  assumed  to  be  0.2833 
pound. 


460 


APPLIED    MECHANICS. 


PLATES  \  INCH  AND  OVER  IN  THICKNESS. 


Width  of  Plate. 

Thickness  of 

Plate. 
Inch. 

Up  to  75 
Inches. 

75  to  100 
Inches. 

Over  100 
Inches. 

Per  Cent. 

Per  Cent. 

Per  Cent. 

i 

IO 

14 

18 

A 

8 

12 

16 

1 

7 

IO 

13 

tk 

6 

8 

10 

1 

5 

7 

9 

A 

4* 

6J 

8i 

^ 

4 

6 

8 

Overf 

3* 

5 

6* 

PLATES  UNDER  £  INCH  IN  THICKNESS. 


Thickness  of 
Plate. 
Inch. 

Width  of  Plate. 

Up  to  50 
Inches. 
Per  Cent. 

50  Inches 
and  Above. 
Per  Cent. 

£     up  to  ^j 

5       '  i     t  t     3 
32                   16 
3      <  (     "1 
16                    1 

10 

7 

Si 

IO 

FINISH. 

1 6.  All  finished  material  shall  be  free  from  injurious  surface  defects 
and  laminations,  and  must  have  a  workmanlike  finish. 

BRANDING. 

17.  Every  finished  piece  of  steel  shall  be  stamped  with  the  melt 
number,  and  each  plate  and  the  coupon  or  test  specimen  cut  from  it 
shall  be  stamped  with  a  separate  identifying  mark  or  number.     Rivet 
steel  may  be  shipped  in  bundles  securely  wired  together  with  the  melt 
number  on  a  metal  tag  attached. 

INSPECTION. 

1 8.  The  inspector,  representing  the  purchaser,  shall  have  all  reason- 
able facilities  afforded  to  him  by  the  manufacturer  to  satisfy  him  that 
the  finished  material  is  furnished  in  accordance  with  these  specifica- 
tions.    All  tests  and  inspections  shall  be  made  at  the  place  of  manu- 
facture, prior  to  shipment. 


STRUCTURAL   STEEL   FOR  BUILDINGS. 


461 


STRUCTURAL  STEEL  FOR  BUILDINGS. 

Adopted  1901. 

PROCESS  OF  MANUFACTURE. 

1.  Steel  may  be  made  by  either  the  open-hearth  or  Bessemer  process. 

CHEMICAL  PROPERTIES. 

2.  Each  of  the  two  classes  of  structural  steel  for  buildings  shall 
not  contain  more  than  o.  10  per  cent  of  phosphorus. 

PHYSICAL  PROPERTIES. 

3.  There  shall  be  two  classes  of  structural  steel  for  buildings, — 
namely,  rivet  steel  and  medium  steel,— which  shall  con-    Classes, 
form  to  the  following  physical  qualities: 

4.  Tensile  Tests. 


Rivet  Steel. 

Medium  Steel. 

Tensile  strength,  pounds  per  square  inch  . 
Yield-point,  in  pounds  per  square  inch,  shall 
not  be  less  than    

50000  to  60000 

*T.  S. 

60000  to  70000 
*T.  S. 

Elongation,  per  cent  in  8  inches  shall  not  be 
less  than  

26 

22 

5.  For  material  less  than  five-sixteenths  inch  (h")  and  more  than 
three-fourths  inch  (}")  in  thickness  the  following  modifica- 
tions shall  be  made  in  the  requirements  for  elongation:      Modifications 

(a)  For  each  increase  of  one-eighth  inch  (£")  in  thick-  f^™*  and" 
ness   above   three-fourths   inch    (f")    a  deduction  of  oneThick  Material- 
per  cent  (i%)  shall  be  made  from  the  specified  elongation. 

(b)  For  each    decrease  of  one-sixteenth   inch    (T&")   in  thickness 
below  five-sixteenths  inch  (&")  a  deduction  of  two  and  one-half  per 
cent  (2^%)  shall  be  made  from  the  specified  elongation. 

(c)  For  pins  the  required  elongation  shall  be  five  per  cent  (5%) 
less  than  that  specified  in  paragraph  No.  4,  as  determined  on  a  test 
specimen  the  centre  of  which  shall  be  one  inch  (i")  from  the  surface. 

6.  The  two  classes  of  structural  steel  for  buildings  shall  conform 
to  the  following  bending  tests ;   and  for  this  purpose  the  test    Bending> 
specimen  shall  be  one  and  one-half  inches  (ij")  wide,  if   Tests- 
possible,  and  for  all  material  three-fourths  ( J")  or  less  in  thickness  the  test 
specimen  shall  be  of  the  same  thickness  as  that  of  the  finished  material 


462  APPLIED    MECHANICS. 

from  which  it  is  cut,  but  for  material  more  than  three-fourths  inch  (f  ") 
thick  the  bending-test  specimen  may  be  one-half  inch  (J")  thick. 
Rivet  rounds  shall  be  tested  of  full  size  as  rolled. 

(d)  Rivet  steel  shall  bend  cold  180°  flat  on  itself  without  fracture 
on  the  outside  of  the  bent  portion. 

(e)  Medium  steel  shall  bend  cold  180°  around  a  diameter  equal 
to  the  thickness  of  the  specimen  tested,  without  fracture  on  the  outside 
of  the  bent  portion. 

TEST  PIECES  AND  METHODS  OF  TESTING. 

7.  The  standard  test  specimen  of  eight-inch  (8")  gauged  length 

shall  be  used  to  determine  the  physical  properties  specified 
men  for  Ten-  in  paragraphs  Nos.  4  and  5.  The  standard  shape  of  the 

test  specimen  for  sheared  plates  shall  be  as  shown  by 
Fig.  2.  (See  page  398.)  For  other  material  the  test  specimen  may  be 
the  same  as  for  sheared  plates  or  it  may  be  planed  or  turned  parallel 
throughout  its  entire  length  and,  in  all  cases  where  possible,  two  oppo- 
site sides  of  the  test  specimen  shall  be  the  rolled  surfaces.  Rivet 
rounds  and  small  rolled  bars  shall  be  tested  of  full  size  as  rolled. 

8.  One   tensile-test    specimen   shall    be    taken   from  the   finished 
Number  of       material  of  each  melt  or  blow  ;    but  in  case  this  develops 
Tensile  Tests.   flaws>  or  breaks  outside  of  the  middle  third  of  its  gauged 
length,  it  may  be  discarded  and  another  test  specimen  substituted 
therefor. 

9.  One  test  specimen  for  bending  shall  be  taken  from  the  finished 

material  of  each  melt  or  blow  as  it  comes  from  the  rolls,  and 


Test 

men  for  for  material  three-fourths  inch  (f  ")  and  less  in  thickness 
this  specimen  shall  have  the  natural  rolled  surface  on  two 
opposite  sides.  The  bending-test  specimen  shall  be  one  and  one- 
half  inches  (ij")  wide,  if  possible;  and  for  material  more  than  three- 
fourths  inch  (I")  thick  the  bending-test  specimen  may  be  one-half 
inch  (I")  thick.  The  sheared  edges  of  bending-test  specimens  may 
be  milled  or  planed. 

Rivet  rounds  shall  be  tested  of  full  size  as  rolled. 

(/)  The  bending  test  may  be  made  by  pressure  or  by  blows. 

10.  Material  which  is  to  be  used  without  annealing  or  further 
Annealed  treatment  shall  be  tested  for  tensile  strength  in  the  con- 
dition  in  which  it  comes  from  the  rolls.  Where  it  is 


STRUCTURAL  STEEL   FOR  BUILDINGS.  463 

impracticable  to  secure  a  test  specimen  from  material  which  has 
been  annealed  or  otherwise  treated,  a  full-sized  section  of  tensile- 
test  specimen  length  shall  be  similarly  treated  before  cutting  the  tensile- 
test  specimen  therefrom. 

11.  For  the  purposes  of  this  specification  the  yield-point  shall  be 
determined  by  the  careful  observaton  of  the  drop  of  the    Yield-point. 
beam  or  halt  in  the  gauge  of  the  testing  machine. 

12.  In  order   to   determine   if  the   material  conforms 

(o  the  chemical  limitations  prescribed  in  paragraph  No.  2    ch?micai°r 
herein,  analysis  shall  be  made  of  drillings  taken  from  a    AnJ 
small  test  ingot. 

VARIATION  IN  WEIGHT. 

13.  The  variation  in  cross  section  or  weight  of  more  than  2\  per 
cent  from  that  specified  will  be  sufficient  cause  for  rejection,  except  in 
the  case  of  sheared  plates,  which  will  be  covered  by  the  following  per- 
missible variations : 

(g)  Plates  12^  pounds  per  square  foot  or  heavier,  up  to  100  inches 
wide,  when  ordered  to  weight,  shall  not  average  more  than  2j  per 
cent  variation  above  or  2\  per  cent  below  the  theoretical  weight. 
When  100  inches  wide,  and  over  5  per  cent  above  or  5  per  cent  below 
the  theoretical  weight. 

(h)  Plates  under  12  J  pounds  per  square  foot,  when  ordered  to 
weight,  shall  not  average  a  greater  variation  than  the  following  : 

Up  to  75  inches  wide,  2^  per  cent  above  or  2\  per  cent  below  the 
theoretical  weight.  Seventy-five  inches  wide  up  to  100  inches  wide,  5 
per  cent  above  or  3  per  cent  below  the  theoretical  weight.  When  100 
inches  wide  and  over,  10  per  cent  above  or  3  per  cent  below  the 
theoretical  weight.  , 

(i)  For  all  plates  ordered  to  gauge,  there  will  be  permitted  an 
average  excess  of  weight  over  that  corresponding  to  the  dimensions 
on  the  order  equal  in  amount  to  that  specified  in  the  following  table : 
TABLE  OF  ALLOWANCES  FOR  OVERWEIGHT  FOR  RECTANGULAR  PLATES 
WHEN  ORDERED  TO  GAUGE. 

Plates  will  be  considered  up  to  gauge  if  measuring  not  over  T^ 
inch  less  than  the  ordered  gauge. 

The  weight  of  i  cubic  inch  of  rolled  steel  is  assumed  to  be  0.2833 
pound. 


464 


APPLIED   MECHANICS. 


PLATES  J  INCH  AND  OVER  IN  THICKNESS. 


Width  of  Plate. 

Plate. 

Inch. 

Up  to  75 

75  to  100 

Over  100 

Inches. 
Per  Cent. 

Inches. 
Per  Cent. 

Inches. 
Per  Cent. 

t 

10 

8 

14 

12 

18 
16 

I 

7 

10 

*3 

Tff 

6 

8 

10 

I 

5 

7 

Q 

TS 

4i 

6| 

8* 

£ 

4 

6 

8 

Over  £ 

3* 

5 

6* 

PLATES  UNDER  \  INCH  IN  THICKNESS. 


Width  of  Plate. 

Plate. 

Inch. 

Up  to  50 
Inches. 

50  Inches 
and  Above. 

Per  Cent. 

Per  Cent. 

\    up  to  & 

10 

15 

&   "    "  & 

8J 

za| 

A  "  "  i 

7 

10 

FINISH. 

14.  Finished  material  must  be  free  from  injurious  seams,  flaws,  or 
cracks,  and  have  a  workmanlike  finish. 

BRANDING. 

15.  Every  finished  piece  of  steel  shall  be  stamped  with  the  melt  or 
blow  number,  except  that  small  pieces  may  be  shipped  in  bundles 
securely  wired  together  with  the  melt  or  blow  number  on  a  metal  tag 
attached. 

INSPECTION. 

1 6.  The  inspector,  representing  the  purchaser,  shall  have  all  reason- 
able facilities  accorded  to  him  by  the  manufacturer  to  satisfy  him  that 
the  finished  material  is  furnished  in  accordance  with  these  specifications. 
All  tests  and  inspections  shall  be  made  at  the  place  of  manufacture, 
prior  to  shipment. 


SPECIFICATIONS  FOR  STEEL  FOR  BRIDGES. 


465 


STRUCTURAL  STEEL  FOR  BRIDGES. 

Adopted   1905. 

1.  Steel   shall   be   made   by   the   open-hearth   process.   Manufacture. 

2.  The  chemical  and  physical  properties  shall  conform  chemical  and 


to  the  following  limits: 

Physical 
Properties. 

Elements  Considered. 

Structural  Steel. 

Rivet  Steel. 

Steel  Castings. 

_          f  Basic    . 

o  04  per  cent 

o  04  per  cent 

Phosphorus  Max.    |  Acid 
Sulphur  Max  

0.08 
0.05         " 

0.04         " 
0.04         " 

0.08 
0.05           " 

Lit  tensile  strength     

Desired 

D  esired 

Not  less  than 

Pounds  per  sq  in    

60,000 

50,000 

6c  ooo 

Elong.:    Min.   per  cent,  in  8  in. 
(Fig.  i)  

f    i,  =500,000* 

1,^00,000 

Uong.:    Min.   per  cent,   in  2  in. 

(Fig  2)   . 

\  Ult.  tens.  str. 

22 

Ult.  tens.  str. 

18 

Character  of  fracture 

Silky 

Silky 

Cold  bend  without  fracture  

1  80°  flat  f 

1  80°  flat  % 

granular. 
00°      d  —  3t 

Retests. 


*  See  par.  n.         \  See  par.  12,  13  and  14.         \  See  par.  15. 

The  yield-point,  as  indicated  by  the  drop  of  beam,  shall  be  recorded 
in  the  test  reports. 

3.  If  the  ultimate  strength  varies  more  than  4,000  Ibs.  from  that 
desired,  a  retest  may  be  made,  at  the  discretion  of  the  inspec- 
tor, on  the  same  gauge,  which,  to  be  acceptable,  shall  be 

within  5,000  Ibs.  of  the  desired  ultimate. 

4.  Chemical  determinations  of  the  percentages  of  carbon,   phos- 
phorus,   sulphur,  and  manganese   shall   be   made   by  the  Chemica,  r^. 
manufacturer  from  a  test  ingot  taken  at  the  time  of  the    terminations. 
pouring  of  each  melt  of  steel  and  a  correct  copy  of  such  analysis  shall 
be  furnished  to  the  engineer  or  his  inspector.     Check  analyses  shall  be 
made  from  finished  material,  if  called  for  by  the  purchaser,  in  which 
case  an  excess  of  25  per  cent  above  the  required  limits  will  be  allowed. 

5.  Specimens  for  tensile  and  bending  tests  for  plates,  shapes,  and 
bars  shall  be  made  by  cutting  coupons  from  the  finished  pjates  shapes 
product,  which  shall  have  both  faces  rolled  and  both  edges  and  Bar*- 
milled  to  the  form  shown  by  Fig.  2,  page  398;  or  with  both  edges 


466  APPLIED   MECHANICS. 

parallel;  or  they  may  be  turned  to  a  diameter  of  f  inch  for  a  length 

of  at  least  9  inches,  with  enlarged  ends. 

Rivets.  6.  Rivet  rods  shall  be  tested  as  rolled. 

7.  Specimens  shall  be  cut  from  the  finished  rolled  or  forged  bar  in 
Pins  and         such  manner  that  the  centre  of  the  specimen  shall  be  i 
Rollers.  incn  from  the  surface  of  the  bar.     The  specimen  for  tensile 
test  shall  be  turned  to  the  form  shown  by  Fig.  i,  page  398.     The 
specimen  for  bending  test  shall  be  i  inch  by  J  inch  in  section. 

8.  The  number  of  tests  will  depend  on  the  character  and  import- 
steel  Cast-       ance  of  the  castings.     Specimens  shall  be  cut  cold  from 
ings.  coupons  moulded  and  cast  on  some  portion  of  one  or  more 
castings  from  each  melt  or  from  the  sink-heads,  if  the  heads  are  of 
sufficient  size.     The  coupon  or  sink-head,  so  used,  shall  be  annealed 
with  the  casting  before  it  is  cut  off.     Test  specimens  to  be"  of  the  form 
prescribed  for  pins  and  rollers. 

9.  Material  which  is  to  be  used  without  annealing  or  further  treat- 
Conditions       ment  shall  be  tested  in  the  condition  in  which  it  comes 
for  Tests.         from  the  rolls.     When  material  is  to  be  annealed  or  other- 
wise treated  before  use,  the  specimens  for  tensile  tests,  representing 
such  material,  shall  be  cut  from  properly  annealed  or  similarly  treated 
short  lengths  of  the  full  section  of  the  bar. 

10.  At  least  one  tensile  and  one  bending  test  shall  be  made  from 
Number  of       each  melt  of  steel  as  rolled.     In  case  steel  differing  f  inch 
Tests.  and  more  in  thickness  is  rolled  from  one  melt,  a  test  shall  be 
made  from  the  thickest  and  thinest  material  rolled. 

11.  For  material  less    than  5-16  inch  and  more  than  f  inch  in 

thickness  the  following  modifications  will  be  allowed  in  the 

Elongation.  .  r          ,  . 

requirements  for  elongation: 

(a)  For  each  1-16  inch  in  thickness  below  5-16  inch,  a  deduction 

of  2^  will  be  allowed  from  the  specified  percentage. 

(b)  For  each  J  inch  in  thickness  above  }  inch,  a  deduction  of  i 

will  be  allowed  from  the  specified  percentage. 

12.  Bending  tests  may  be  made  by  pressure  or' by  blows.     Plates, 
Bendin  shapes,  and  bars  less  than  i  inch  thick  shall  bend  as  called 
Tests.             for  m  paragraph  2. 

13.  Full-sized  material  for  eye-bars  and  other  steel  i  inch  thick 


SPECIFICATIONS  FOR  STEEL  FOR  BRIDGES.  467 

and  over,  tested  as  rolled,  shall  bend  cold   180°  around    F  „  •   d 
a  pin  the  diameter  of  which  is  equal  to  twice  the  thickness    Bends 
of  the  bar,  without  a  fracture  on  the  outside  of  bend. 

14.  Angles  f  inch  and  less  in  thickness  shall  open  flat,  and  angles 
J  inch  and  less  in -thickness  shall  bend  shut,  cold,  under    Testson 
blows  of  a  hammer,  without  sign  of  fracture.      This  test    Angles, 
will  be  made  only  when  required  by  the  inspector. 

15.  Rivet  steel,  when  nicked  and  bent  around  a  bar  of  the  same 
diameter  as  the  rivet  rod,  shall  give  a  gradual  break  and  a    Tests  on 
fine,  silky,  uniform  fracture.  Rivet  stee!- 

1 6.  Finished  material  shall  be  free  from  injurious  seams,   flaws, 
cracks,  defective  edges,  or  other  defects,  and  have  a  smooth 
uniform,   workmanlike  finish.     Plates  36  inches  in  width        IS  * 
and  under  shall  have  rolled  edges. 

17.  Every  finished  piece  of  steel  shall  have  the  melt  number  and 
the  name  of  the  manufacturer  stamped  or  rolled  upon  it.    Markin 
Steel  for  pins  and  rollers  shall  be  stamped  on  the  end. 

Rivet  and  lattice  steel  and  other  small  parts  may  be  bundled  with  the 
above  marks  on  an  attached  metal  tag. 

1 8.  Material  which,  subsequent  to  the  above  tests  at  the  mills  and 
its    acceptance    there,    develops    weak    spots,    brittleness, 

cracks  or  other  imperfections,  or  is  found  to  have  injurious      ejecl 
defects,  will  be  rejected  at  the  shop  and  shall  be  replaced  by  the  manu- 
facturer at  his  own  cost. 

19.  A  variation  in  cross-section  or  weight  of  each  piece  of  steel 
of  more  than  2\  per  cent  from  that  specified  will  be  suffi-    pei.missjb|e 
cient  cause  for  rejection,  except  in  case  of  sheared  plates,    Variations, 
which  will  be  covered  by  the  following  permissible  variations,  which 
are  to  apply  to  single  plates. 

WHEN  ORDERED  TO  WEIGHT. 

20.  Plates  12^  pounds  per  square  foot  or  heavier:  Variations? 

(a)  Up  to  100  inches  wide,  2\  per  cent  above  or  below  the  pre- 

scribed weight. 

(b)  One  hundred  inches  wide  and  over,  5  per  cent  above  or  below. 

21.  Plates  under  i2\  pounds  per  square  foot: 

(a)  Up  to  75  inches  wide,  2j  per  cent  above  or  below. 


468 


APPLIED    MECHANICS. 


(b)  Seventy-five  inches  and  up  to  100  inches  wide,  5  per  cent  above 

or  3  per  cent  below. 

(c)  One    hundred  inches  wide  and  over,  10  per  cent  above  or  3 

per  cent  below. 


Permissible 
Variations. 


WHEN  ORDERED  TO  GAUGE. 

22.  Plates  will  be  accepted  if  they  measure  not  more 
than  o.oi  inch  below  the  ordered  thickness. 

23.  An  excess  over  the  nominal  weight  corresponding  to  the  dimen- 
sions on  the  order,  will  be  allowed  for  each  plate,  if  not  more  than  that 
shown  in  the  following  tables,  one  cubic  inch  of  rolled  steel  being 
assumed  to  weigh  0.2833  pound. 

24.  Plates  i  inch  and  over  in  thickness. 


Thickness 
Ordered. 

Nominal 
Weights. 

Width  of  Plate. 

Up  to  75". 

75"  and  up 
to  too". 

ioo"and  up 
to  115'. 

Over  1  1  5". 

1—4  in 
5-16 
3-8 
7-16 

1-2 
9~l6 

5-8 
Over  5-8     ' 

ch. 

10.20    It 

12.75 

15-30 
17-85 

20.40 

22.95 

25-50 

)S. 

10    p 
8 

6 

& 

4 
3* 

er  ce 

nt. 

14    p 
12 
IO 

8 

ij 

5 

er  ce 

nt. 

18    p 
i6 

13 

IO 

!> 

6* 

er  ce 

nt. 

17  per  cent. 
13    ' 

12      ' 

ii    ' 

10      ' 

9    ' 

25.  Plates  under  J  inch  in  thickness. 


Thickness  Ordered. 

Nominal  Weights. 
Pounds  per 
Square  Feet. 

Width  of  Plate. 

Up  to  50". 

50"  and  up  to 

70". 

Over  70", 

1-8"  up  to  5-32" 
5-32"  "  3-i6" 
3-16'"  "  1-4" 

5.10  to     6.37 
6-37  "      7-65 

7.65    "     IO.2O 

10    per  cent. 
SJ  "      « 

7     "      " 

15    per  cent. 

12*    "         " 
10        "         " 

20  per  cent. 
17  "      " 

15   "      " 

26.  The  purchaser  shall  be  furnished  complete  copies  of  mill  orders 
ins  ction  and  no  material  shall  be  rolled,  nor  work  done,  before  the 
and  Testing,  purchaser  has  been  notified  where  the  orders  have  been 
placed,  so  that  he  may  arrange  for  the  inspection. 


STRUCTURAL   STEEL   FOR   SHIPS. 


469 


27.  The  manufacturer  shall  furnish  all  facilities  for  inspecting  and 
testing  the  weight  and  quality  of  all  material  at  the  mill  where  it  is 
manufactured.     He  shall  furnish  a  suitable  testing  machine  for  testing 
the  specimens,  as  well  as  prepare  the  pieces  for  the  machine,  free  of  cost. 

28.  When  an  inspector  is  furnished  by  the  purchaser  to  inspect 
material  at  the  mills,  he  shall  have  full  access,  at  all  times,  to  all  parts 
of  mills  where  material  to  be  inspected  by  him  is  being  manufactured. 

STRUCTURAL   STEEL   FOR   SHIPS. 

Adopted  1901  for  bridges  and  ships.     Restricted  to  ships,  1905. 

PROCESS  OF  MANUFACTURE. 

1.  Steel  shall  be  made  by  the  open-hearth  process. 

CHEMICAL  PROPERTIES. 

2.  Each  of  the  three  classes  of  structural  steel  for  ships  shall  con- 
form to  the  following  limits  in  chemical  composition : 


Steel  Made  by 
the  Acid 
Process. 
Per  Cent. 

Steel  Made  by 
the  Basic 
Process. 
Per  Cent. 

Phosphorus  shall  not  exceed  . 
Sulphur 

O.o8 
0.06 

O.o6 
0.06 

PHYSICAL  PROPERTIES. 

3.  There  shall  be   three   classes   of  structural   steel   for  ships, — 
namely,  rivet  steel,  soft  steel,  and   medium  steel, — which 

shall  conform  to  the  following  physical  qualities: 

4.  Tensile  Tests. 


Classes. 


Rivet  Steel. 

Soft  Steel. 

Medium  Steel. 

Tensile    strength,    pounds 

per  square  inch    . 

50000  to  60000 

52000  to  62000 

60000  to  70000 

Yield  -point,  in  pounds  per 

square  inch  shall  not  be 

less  than    

IT.S. 

JT.  S. 

JT.S. 

Elongation,  per  cent  in   8 

inches  shall  not  be  less 

than     

26 

25 

22 

4/0  APPLIED    MECHANICS. 


5.  For  material  less  than  five-sixteenths  inch  (&")  and  more  than 

three-fourths  inch  (}")  in  thickness  the  following  modifi- 
Modifications    cations  shall  be  made  in  the  requirements  for  elongation : 
for  Thin  and  (#)   For  each  increase  of  one-eighth  inch  ( J")  in  thickness 

'above  three-fourths  inch  (}")  a  deduction  of  one  per  cent 
(i%)  shall  be  made  from  the  specified  elongation. 

(b)  For  each    decrease   of  one-sixteenth   inch   (TS")   in  -thickness 
below  five-sixteenths  inch  (&")  a  deduction  of  two  and  one-half  per 
cent  (2  J%)  shall  be  made  from  the  specified  elongation. 

(c)  For  pins  made  from  any  of  the  three  classes  of  steel  the  required 
elongation  shall  be  five  per  cent  (5%)  less  than  that  specified  in  para- 
gaph  No.  4,  as  determined  on  a  test  specimen,  the  center  of  which  shall 
be  one  inch  (i")  from  the  surface. 

6.  Eye-bars  shall  be  of  medium  steel.     Full-sized  tests  shall  show 
Tensile  Tests    I2i  Per  cent  elongation  in  fifteen  feet  of  the  body  of  the 
of  Eye-bars,     eye-bar,  and  the  tensile  strength  shall  not  be  less  than 
55,000  pounds  per  square  inch.     Eye-bars  shall  be  required  to  break 
in  the  body;  but,  should  an  eye-bar  break  in  the  head,  and  show  twelve 
and  one-half  per  cent  (12^%)  elongation  in  fifteen  feet  and  the  tensile 
strength  specified,  it  shall  not  be  cause  for  rejection,  provided  that  not 
more  than  one-third  (J)  of  the  total  number  of  eye-bars  tested  break 
in  the  head. 

7.  The  three  classes  of  structural  steel  for  ships  shall  conform 
Bendin  *°    ^e    f°M°wmg    bending    tests;    and    for    this    purpose 
Tests.  the  test  specimen  shall  be  one  and  one-half  inches  wide, 
if  possible,  and  for  all  material  three-fourths  inch  (f ")  or  less  in  thick- 
ness the  test  specimen  shall  be  of  the  same  thickness  as  that  of  the 
finished  material  from  which  it  is  cut,  but  for  material  more  than 
three-fourths  inch  (f")  thick  the  bending-test  specimen  may  be  one- 
half  inch  (£")  thick. 

Rivet  rounds  shall  be  tested  of  full  size  as  rolled. 

(d)  Rivet  steel  shall  bend  cold  180°  flat  on  itself  without  fracture 
on  the  outside  of  the  bent  portion. 

(e)  Soft  steel  shall  bend  cold  180°  flat  on  itself  without  fracture  on 
the  outside  of  the  bent  portion. 

(/)  Medium  steel  shall  bend  cold  180°  around  a  diameter  equal  to 
the  thickness  of  the  specimen  tested,  without  fracture  on  the  outside  of 
the  bent  portion. 


STRUCTURAL   STEEL   FOR   SHIPS.  47  1 

TEST  PIECES  AND  METHODS  OF  TESTING. 

8.  The  standard  test  specimen  of  eight  inch  (8")  gauged  length  shall 
be  used  to  determine  the  physical  properties  specified  in 


paragraphs  Nos.  4  and  5.  The  standard  shape  of  the  test  men 
specimen  for  sheared  plates  shall  be  as  shown  by  Fig.  2, 
page  398.  For  other  material  the  test  specimen  may  be  the  same  as 
for  sheared  plates,  or  it  may  be  planed  or  turned  parallel  throughout 
its  entire  length;  and,  in  all  cases  where  possible,  two  opposite  sides 
of  the  test  specimens  shall  be  the  rolled  surfaces.  Rivet  rounds  and 
small  rolled  bars  shall  be  tested  of  full  size  as  rolled. 

9.  One  tensile-test  specimen  shall  be  taken  from  the  finished  material 
of  each  melt;   but  in   case  this  develops  flaws,  or  breaks    Numb^ 
outside  of  the  middle  third  of  its  gauged  length,  it  may   Tensile  Tests. 
be  discarded,  and  another  test  specimen  substituted  therefor. 

10.  One  test  specimen  for  bending  shall  be  taken  from  the  finished 
material  of  each  melt  as  it  comes  from  the  rolls,  and  for 
material   three-fourths    inch    (f  ")    and   less    in   thickness    nSns  forCI" 
this   specimen  shall  have  the   natural  rolled   surface  on    Bendmg- 
two  opposite  sides.     The  bending-test  specimen  shall  be  one  and  one 
half  inches  (i  J")  wide,  if  possible,  and  for  material  more  than  three- 
fourths  inch   (I")  thick  the  bending-test  specimen  may  be  one-half 
inch  (J")  thick.      The  sheared  edges  of  bending-test  specimens  may 
be  milled  or  planed. 

(g)  The  bending  test  may  be  made  by  pressure  or  by  blows. 

11.  Material  which  is  to  be  used  without  annealing  or  further 
treatment  shall  be  tested  for  tensile  strength  in  the  con- 

dition in  which  it  comes  from  the  rolls.     Where  it  is  imprac-    Test  Speci- 
ticable   to   secure  a  test   specimen   from   material   which 
has  been  annealed  or  otherwise  treated,  a  full-sized  section  of  tensile 
test,  specimen  length,  shall  be  similarly  treated  before  cutting  the 
tensile-test  specimen  therefrom. 

12.  For  the  purpose  of  this  specification  the  yield-point  shall  be 
determined  by  the  careful  observation  of  the  drop  of  the 

beam  or  halt  in  the  gauge  of  the  testing  machine. 

13.  In  order  to  determine  if  the  material  conforms  to 

the  chemical  limitations  prescribed   in  paragraph  No.   2    Ch?mica?r 
herein,  analysis  shall  be  made  of  drillings  taken  from  a 
small  test  ingot. 


472 


APPLIED    MECHANICS. 


VARIATION  IN  WEIGHT. 

14.  The  variation  in  cross  section  or  weight  of  more  than  2j  per 
cent  from  that  specified  will  be  sufficient  cause  for  rejection,  except  in 
the  case  of  sheared  plates,  which  will  be  covered  by  the  following  per- 
missible variations: 

(h)  Plates  i2j  pounds  per  square  foot  or  heavier,  up  to  100  inches 
wide,  when  ordered  to  weight,  shall  not  average  more  than  2  J  per  cent 
variation  above  or  2^  per  cent  below  the  theoretical  weight.  When 
100  inches  wide  and  over,  5  per  cent  above  or  5  per  cent  below  the 
theoretical  weight. 

(i)  Plates  under  12^  pounds  per  square  foot,  when  ordered  to  weight, 
shall  not  average  a  greater  variation  than  the  following: 

Up  to  75  inches  wide,  2j  per  cent  above  or  2\  per  cent  below  the 
theoretical  weight.  75  inches  wide  up  to  100  inches  wide,  5  per  cent 
above  or  3  per  cent  below  the  theoretical  weight.  When  100  inches  wide 
and  over,  10  per  cent  above  or  3  per  cent  below  the  theoretical  weight. 

(j)  For  all  plates  ordered  to  gauge  there  will  be  permitted  an  average 
excess  of  weight  over  that  corresponding  to  the  dimensions  on  the  order 
equal  in  amount  to  that  specified  in  the  following  table : 

TABLE  OF  ALLOWANCES  FOR  OVERWEIGHT  FOR  RECTANGULAR  PLATES 

WHEN   ORDERED   TO    GAUGE. 

Plates  will  be  considered  up  to  gauge  if  measuring  not  over  T^-¥ 
inch  less  than  the  ordered  gauge. 

The  weight  of  i  cubic  inch  of  rolled  steel  is  assumed  to  be  0.2833 

pound. 

PLATE  £  INCH  AND  OVER  IN  THICKNESS. 


Width  of  Plate. 

Plate. 
Inch. 

Up  to  75 
Inches. 

75  to  TOO 
Inches. 

Over  100 
Inches. 

Per  Cent. 

Per  Cent. 

Per  Cent. 

£ 

10 

14 

18 

& 

8 

12 

16 

| 

7 

10 

13 

A 

6 

8 

IO 

i 

5 

7 

Q 

& 

4* 

6* 

8i 

f 

4 

6 

8 

Overf 

3i 

5 

6* 

STEEL   AXLES. 


473 


PLATES  UNDER  \  INCH  IN  THICKNESS. 


Thickness  of 
Plate. 
Inch. 

Width  of  Plate. 

Up  to  50 
Inches. 
Per  Cent. 

50  Inches 
and  Above. 
Per  Cent. 

\    up  to  & 

A  "  "A 
&  "  "i* 

10 

8* 

7 

11, 

10 

FINISH. 

15.  Finished  material  must  be  free  from  injurious  seams,  flaws,  or 
cracks,  and  have  a  workmanlike  finish. 

BRANDING. 

1 6.  Every  finished  piece  of  steel  shall  be  stamped  with  the  melt 
number,  and  steel  for  pins  shall  have  the  melt  number  stamped  on  the 
ends.     Rivets  and  lacing  steel,  and  small  pieces  for  pin  plates  and 
stiffeners,  may  be  shipped  in  bundles,  securely  wired  together,  with  the 
melt  number  on  a  metal  tag  attached. 

INSPECTION. 

17.  The  inspector,  representing  the  purchaser,  shall  have  all  reason- 
able facilities  afforded  to  him  by  the  manufacturer  to  satisfy  him  that 
the  finished  material  is  furnished  in  accordance  with  these  specifica- 
tions.    All  tests  and  inspections  shall  be  made  at  the  place  of  manu- 
facture, prior  to  shipment. 

STEEL  AXLES. 

Adopted  1901.     Modified  1905. 

PROCESS  OF  MANUFACTURE. 

1.  Steel  for  axles  shall  be  made  by  the  open-hearth  process. 

CHEMICAL  PROPERTIES. 

2.  There  shall  be  three  classes  of  steel  axles,  which  shall  conform 
to  the  following  limits  in  chemical  composition: 


474 


APPLIED   MECHANICS. 


Car  and 
Tender-truck 
Axles. 

Per  Cent. 

Driving  and 
Engine  -truck 
Axles. 
(Carbon  Steel.) 
Per  Cent. 

Driving-wheel 
Axles. 
(Nickel-steel.) 

Per  Cent. 

Phosphorus  shall  not  exceed  

0.06 

O.o6 

O   O4 

Sulphur           "      "        " 

o  06 

c  06 

o  04 

Manganese      "      "        "      
Nickel  

0.60 

3O  to  4    O 

PHYSICAL  PROPERTIES. 

3.  For  car  and  tender- truck  axles,  no  tensile  test  shall 
be  required. 

4.  The  minimum  physical  qualities  required  in  the  two  classes  of 
driving-wheel  axles  shall  be  as  follows : 


Tensile  Tests. 


Driving  and 
Engine-truck 
Axles. 
(Carbon  Steel.) 

Driving  and 
Engine-truck 
Axles. 
(Nickel  steel.) 

Tensile  strength  pounds  per  square  inch            .... 

80,000 
40,000 

20 

25 

80,000 
50,000 
25 
45 

Yield-point    pounds  per  square  inch     

Elongation  per  cent  in  two  inches        

Contraction  of  area  per  cent  .       • 

5.  One  axle  selected  from  each  melt,  when  tested  by  the  drop  test 
described  in  paragraph  No.  9,  shall  stand  the  number  of 
blows  at  the  height  specified  in  the  following  table  without 
rupture  and  without  exceeding,  as  the  result  of  the  first  blow,  the  deflec- 
tion given.     Any  melt  failing  to  meet  these  requirements  will  be  rejected. 


Diameter  of 
Axle  at  Center. 
Inches. 

Number  of 
Blows. 

Height  of 
Drop. 
Feet 

Deflection. 
Inches. 

4i 

5 

24 

81 

4§ 

5 

26 

8i 

4^T5 

•  5 

28} 

81 

4i 

5 

31 

,      8 

4f 

5 

34 

8 

5l 

5 

43 

7, 

5i 

7 

43 

Si       . 

6.  Carbon-steel  and   nickel-steel   driving-wheel  axis   shall  not 
subject  to  the  above  drop  test. 


be 


STEEL   AXLES.  475 


TEST  PIECES  AND  METHODS  OF  TESTING. 


7.  The  standard  test  specimen  one-half  inch  (J")   diameter  and 
two  inch  (2")  gauged  length  shall  be  used  to  determine 

the  physical  properties  specified  in  paragraph  No.  4.     It    men  fo?Ten- 

,  -,-,.  /c,  0  \  sile  Tests. 

is  shown  in  rig.  i.     (See  p.  398.) 

8.  For  driving  and  engine-truck  axles  one  longitudinal  test  specimen 
shall  be  cut  from  one  axle  of  each  melt.     The  center  of  Numberand 
this  test  specimen  shall  be  half-way  between  the  center  T^/e  §  ofci_ 
and  outside  of  the  axle.  mens- 

9.  The  points  of  supports  on  which  the  axle  rests  during  tests  must 
be  three  feet  apart  from  center  to  center;    the  tup  must    DropTest 
weigh   1,640  pounds;    the  anvil,   which   is  supported  on    Described. 
springs,  must  weigh  17,500  pounds;   it  must  be  free  to  move  in  a  ver- 
tical direction;  the  springs  upon  which  it  rests  must  be  twelve  in  number, 
of  the  kind  described  on  drawing;   and  the  radius  of  supports  and  of 
the  striking  face  on  the  tup  in  the  direction  of  the  axis  of  the  axle  must 
be  five  (5)  inches.     When  an  axle  is  tested,  it  must  be  so  placed  in  the 
machine  that  the  tup  will  strike  it  midway  between  the  ends;   and  it 
must  be  turned  over  after  the  first  and  third  blows,  and,  when  required, 
after  the  fifth  blow.     To  measure  the  deflection  after  the  first  blow, 
prepare  a  straight  edge  as  long  as  the  axle,  by  reinforcing  it  on  one  side, 
equally  at  each  end,  so  that,  when  it  is  laid  on  the  axle,  the  reinforced 
parts  will  rest  on  the  collars  or  ends  of  the  axle,  and  the  balance  of  the 
straight  edge  not  touch  the  axle  at  any  place.     Next  place  the  axle  in 
position  for  test,  lay  the  straight  edge  on  it,  and  measure  the  distance 
from  the  straight  edge  to  the  axle  at  the  middle  point  of  the  latter. 
Then,  after  the  first  blow,  place  the  straight  edge  on  the  now  bent  axle 
in  the  same  manner  as  before,  and  measure  the  distance  from  it  to  that 
side  of  the  axle  next  to  the  straight  edge  at  the  point  farthest  away  from 
the  latter.     The  difference  between  the  two  measurements  is  the  de- 
flection.    The  report  of  the  drop  test  shall  state  the  atmospheric  tem- 
perature at  the  time  the  tests  were  made. 

10.  The  yield-point  specified  in  paragraph  No.  4  shall  be  determined 
by  the  careful  observation  of  the  drop  of  the  beam  or  halt 

.J  .  Yield-point. 

in  the  gauge  of  the  testing  machine. 


4/6 


APPLIED   MECHANICS. 


11.  Turnings  from  the  tensile-test  specimen  of  driving  and  engine- 

truck  axles,  or  drillings  taken  midway  between  the  center 
Sample  for  and  outside  of  car,  engine,  and  tender-truck  axles,  or 
Analysis!  drillings  from  the  small  test  ingot,  if  preferred  by  the 

inspector,  shall  be  used  to  determine  whether  the  melt  is 
within  the  limits  of  chemical  composition  specified  in  paragraph  No.  2. 

FINISH. 

12.  Axles  shall  conform  in  sizes,  shapes,  and  limiting  weights  to  the 
requirements  given  on  the  order  or  print  sent  with  it.     They  shall  be 
made  and  finished  in  a  workmanlike  manner,  and  shall  be  free  from 
all  injurious  cracks,  seams,  or  flaws.     In  centering,  sixty-  (60)  degree 
centers  must  be  used,  with  clearance  given  at  the  point  to  avoid  dulling 
the  shop  lathe  centers. 

BRANDING. 

13.  Each  axle  shall  be  legibly  stamped  with  the  melt  number  and 
initials  of  the  maker  at  the  places  marked  on  the  print  or  indicated  by 
the  inspector. 

INSPECTION. 

14.  The  inspector,  representing  the  purchaser,  shall  have  all  reason- 
able facilities  afforded  to  him  by  the  manufacturer  to  satisfy  him  that 
the  finished  material  is  furnished  in  accordance  with  these  specifications. 
All  tests  and  inspections  shall  be  made  at  the  place  of  manufacture, 
prior  to  shipment. 

STEEL  TIRES. 
Adopted  1901. 

PROCESS  OF  MANUFACTURE. 

1.  Steel  for  tires  may  be  made  by  either  the  open-hearth  or  crucible 

process. 

CHEMICAL  PROPERTIES. 

2.  There  will  be  three  classes  of  steel  tires  which  shall  conform 
to  the  following  limits  in  chemical  composition: 


Passenger 
Engines. 

Per  Cent. 

Freight-engine 
and 
Car-wheels. 
Per  Cent. 

Switching- 
engines. 

Per  Cent. 

Manganese  shall  not  exceed  
Silicon  shall  not  be  less  than  

0.80 
o.  20 

0.8o 
O.2O 

0.8o 

O.2O 

Phosphorus  shall  not  exceed  

0.05 

O    Os 

O.O5 
O    O? 

O.O5 

O    O  ^ 

STEEL    TIRES. 


477 


PHYSICAL  PROPERTIES. 

3.  The  minimum  physical  qualities  required  in  each  of 
the  three  classes  of  steel  tires  shall  be  as  follows : 


Tensile  Tests. 


Passenger- 
engines. 

Freight- 
engine  and 
Car-wheels. 

Switching- 
engines. 

Tensile  strength,  pounds  per  square  inch. 
Elongation,  per  cent  in  two  inches      .... 

100,000 

12 

110,000 
IO 

120,000 

g 

Drop  Test. 


4.  In  the  event  of  the  contract  calling  for  a  drop  test,  a  test  tire 
from  each  melt  will  be  furnished  at  the  purchaser's  expense, 
provided  it  meets  the  requirements.     This  test  tire  shall 

stand  the  drop  test  described  in  paragraph  No.  7,  without  breaking  or 
cracking,  and  shall  show  a  minimum  deflection  equal  to  D2-r- 
(4oT2+2D),  the  letter  "D"  being  internal  diameter  and  the  letter 
"T  "  thickness  of  tire  at  center  of  tread. 

TEST  PIECES  AND  METHODS  OF  TESTING. 

5.  The  standard  turned  test  specimen,  one-half  inch  (J")  diameter 
and  two  inch  (2")  gauged  length,  shall  be  used  to  determine   Test  Speci- 
the  physical  properties  specified  in  paragraph  No.  3.     It   {ensile  Tests. 
is  shown  in  Fig.  i.     (See  p.  398.) 

6.  When  the  drop  test  is  specified,  this  test  specimen  shall  be  cut  cold 
from  the  tested  tire  at  the  point  least  affected  by  the  drop    Location  of 
test.     If  the  diameter  of  the  tire  is  such  that 'the  whole  "SSSf Speci" 
circumference  of  the  tire  is  seriously  affected  by  the  drop  test,  or  if  no 
drop  test  is  required,  the  test  specimen  shall  be  forged  from  a  test  ingot 
cast  when  pouring  the  melt,  the  test  ingot  receiving,  as  nearly  as  pos- 
sible, the  same  proportion  of  reduction  as  the  ingots  from  which  the 
tires  are  made. 

7.  The  test  tire  shall  be  placed  vertically  under  the  drop  in  a  run- 
ning position  on  solid  foundation  of  at  least  ten  tons  in    Dro  Test 
weight  and  subjected  to  successive  blows  from  a  tup  weigh-    Described, 
ing  2,240  pounds,  falling  from  increasing  heights  until  the  required 
deflection  is  obtained. 

8.  Turnings  from  the  tensile  specimen,  or  drillings  from  the  small 
test  ingot,  or  turnings  from  the  tire,  if  preferred  by  the 
inspector,  shall  be  used  to  determine  whether  the  melt  is    chemical* 
within   the   limits   of   chemical   composition   specified    in    ' 
paragraph  No.  2. 


478 


APPLIED    MECHANICS. 


FINISH. 

9.  All  tires  shall  be  free  from  cracks,  flaws,  or  other  injurious  im- 
perfections, and  shall  conform  to  dimensions  shown  on  drawings  fur- 
nished t>y  the  purchaser. 

BRANDING. 

10.  Tires  shall  be  stamped  with  the  maker's  brand  and  number  in 
such  a  manner  that  each  individual  tire  may  be  identified. 

INSPECTION. 

11.  The  inspector  representing  the  purchaser  shall  have  all  reason- 
able facilities  afforded  to  him  by  the  manufacturer  to  satisfy  him  that 
the  finished  material  is  furnished  in  accordance  with  these  specifications. 
All  tests  and  inspections  shall  be  made  at  the  place  of  manufacture, 
prior  to  shipment. 

STEEL  RAILS. 

Adopted  1901. 

PROCESS  OF  MANUFACTURE. 

1.  (a)  Steel  may  be  made  by  the  Bessemer  or  open-hearth  process. 

(b)  The  entire  process  of  manufacture  and  testing  shall  be  in  accord- 
ance with  the  best  standard  current  practice,  and  special  care  shall  be 
taken  to  conform  to  the  following  instructions : 

(c)  Ingots  shall  be  kept  in  a  vertical  position  in  pit  heating  furnaces. 

(d)  No  bled  ingots  shall  be  used. 

(e)  Sufficient  material  shall  be  discarded  from  the  top  of  the  ingots 
to  insure  sound  rails. 

CHEMICAL  PROPERTIES. 

2.  Rails  of  the  various  weights  per  yard  specified  below  shall  con- 
form to  the  following  limits  in  chemical  composition: 


50  to  59  + 
Pounds. 
Per  Cent. 

60  to  69  + 
Pounds. 
Per  Cent. 

70  to  79  + 
Pounds. 
Per  Cent. 

80  to  89  + 
Pounds. 
Per  Cent. 

90  to  100 
Pounds. 
Per  Cent. 

Carbon                .... 

o.  3S—  0.45 

0  .  38-0  .  48 

o  .  40—0  .  so 

0.43—0   S3 

O    4S—  O    SS 

Phosphorus  shall  not 
exceed        

O.  IO 

O.  IO 

O.  IO 

O    IO 

•'•IO    uOo 
O    IO 

Silicon  shall  not  ex- 
ceed 

o  20 

o  20 

o  20 

o  20 

M^anganese 

o  70—1  oo 

o  .  70—1  .  oo 

O    7S—  I   OS 

o  80—  i  10 

o  80—  i  10 

STEEL   RAILS. 


479 


PHYSICAL  PROPERTIES. 
3.  One  drop  test  shall  be  made  on  a  piece  of  rail  not  more  than  six 


feet  long,  selected  from  every  fifth  blow  of  steel.  The  rail 
shall  be  placed  head  upwards  on  the  supports,  and  the 
various  sections  shall  be  subjected  to  the  following  impact  tests: 


Drop  Test. 


Weight  of  Rail. 
Pounds  per  Yard. 

Height  of 
Drop. 
Feet. 

45  to  and  including 

55"" 

15 

More  than 

55     " 

65.... 

16 

<  <        <  < 

55 

75-  - 

17 

«        •< 

75     " 

85.... 

18 

85     " 

100.  .  .  . 

J9 

If  any  rail  break  when  subjected  to  the  drop  test,  two  additional  tests 
will  be  made  of  other  rails  from  the  same  blow  of  steel,  and,  if  either  of 
these  latter  tests  fail,  all  the  rails  of  the  blow  which  they  represent  will  be 
rejected ;  but,  if  both  of  these  additional  test  pieces  meet  the  require- 
ments, all  the  rails  of  the  blow  which  they  represent  will  be  accepted. 
If  the  rails  from  the  tested  blow  shall  be  rejected  for  failure  to  meet  the 
requirements  of  the  drop  test,  as  above  specified,  two  other  rails  will 
be  subjected  to  the  same  tests,  one  from  the  blow  next  preceding,  and 
one  from  the  blow  next  succeeding  the  rejected  blow.  In  case  the 
first  test  taken  from  the  preceding  or  succeeding  blow  shall  fail,  two 
additional  tests  shall  be  taken  from  the  same  blow  of  steel,  the  accept- 
ance or  rejection  of  which  shall  also  be  determined  as  specified  above; 
and,  if  the  rails  of  the  preceding  or  succeeding  blow  shall  be  rejected, 
similar  tests  may  be  taken  from  the  previous  or  following  blows,  as  the 
case  may  be,  until  the  entire  group  of  five  blows  is  tested,  if  necessary. 

The  acceptance  or  rejection  of  all  the  rails  from  any  blow  will 
depend  upon  the  result  of  the  tests  thereof. 

TEST  PIECES  AND  METHODS  OF  TESTING. 

4.  The  drop-test  machine  shall  have  a  tup  of  two  thousand  (2,000) 
pounds  weight,  the  striking  face  of  which  shall  have  a  Drop_testing 
radius  of  not  more  than  five  inches  (5"),  and  the  test  rail  Machine. 
shall  be  placed  head  upwards  on  solid  supports  three  test  (3')  apart. 
The  anvil-block  shall  weigh  at  least  twenty  thousand  (20,000)  pounds, 
and  the  supports  shall  be  a  part  of,  or  firmly  secured  to,  the  anvil. 


480  APPLIED    MECHANICS. 

The  report  of  the  drop  test  shall  state  the  atmospheric  temperature  at 
the  time  the  tests  were  made. 

5.  The  manufacturer  shall  furnish  the  inspector  daily  with  carbon 

determinations  of  each  blow,  and  a  complete  chemical 
ChTm!ca?r  ^analysis  every  twenty-four  hours,  representing  the  average 
Analysis.  of  the  Qther  elements  contained  in  the  steel.  These  analy- 
ses shall  be  made  on  drillings  taken  from  a  small  test  ingot. 

FINISH. 

6.  Unless  otherwise  specified,  the  section  of  rail  shall  be  the  Amer- 

ican Standard,  recommended  by  the  American  Society 
of  Civil  Engineers,  and  shall  conform,  as  accurately  as 
possible,  to  the  templet  furnished  by  the  railroad  company,  consistent 
with  paragraph  No.  7,  relative  to  specified  weight.  A  variation  in 
height  of  one-sixty-fourth  of  an  inch  (TV)  less  and  one-thirty-second 
of  an  inch  (fa")  greater  than  the  specified  height  will  be  permitted.  A 
perfect  fit  of  the  splice-bars,  however,  shall  be  maintained  at  all  times. 

7.  The  weight  of  the  rails  shall  be  maintained  as  nearly  as  possible, 

after  complying  with  paragraph  No.  6,  to  that  specified  in 
contract.     A  variation  of  one-half  of  one  per  cent  (4%)  for 

an  entire  order  will  be  allowed.     Rails  shall  be  accepted  and  paid  for 

according  to  actual  weights. 

8.  The  standard  length  of  rails  shall  be  thirty  feet  (30').     Ten  pe 

cent  (10%)  of  the  entire  order  will  be  accepted  in  shorter 
lengths,  varying  by  even  feet  down  to  twenty-four  feet 

(24').     A  variation  of  one-fourth  of  an  inch  (J")  in  length  from  that 

specified  will  be  allowed. 

9.  Circular  holes  for  splice-bars  shall  be  drilled  in  accordance  with 

the  specifications  of  the  purchaser.     The  holes  shall  ac- 
curately conform  to  the  drawing  and  dimensions  furnished 
in  every  respect,  and  must  be  free  from  burrs. 

10.  Rails  shall  be  straightened  while  cold,  smooth  on  head,  sawed 

square  at  ends,  and  prior  to  shipment  shall  have  the  burr 
occasioned  by  the  saw-cutting  removed,  and  the  ends  made 

clean.     No.  i  rails  shall  be  free  from  injurious  defects  and  flaws  of  all 

kinds. 

BRANDING. 

11.  The  name  of  the  maker,  the  month  and  year  of  manufacture, 


STEEL    SPLICE-BARS.  481 

shall  be  rolled  in  raised  letters  on  the  side  of  the  web,  and  the  number 
of  the  blow  shall  be  stamped  on  each  rail. 

INSPECTION. 

12.  The  inspector,  representing  the  purchaser,  shall  have  all  reason- 
able facilities  afforded  to  him  by  the  manufacturer  to  satisfy  him  that 
the  finished  material  is  furnished  in  accordance  with  these  specifications. 
All  tests  and  inspections  shall  be  made  at  the  place  of  manufacture, 
prior  to  shipment. 

No.  2  RAILS. 

13.  Rails  that  possess  any  injurious  physical  defects,   or  which 
for  any  other  cause  are  not  suitable  for  first  quality,  or  No.  i  rails, 
shall  be  considered  as  No.  2  rails,  provided,  however,  that  rails  which 
contain  any  physical  defects  which  seriously  impair  their  strength  shall 
be  rejected.     The  ends  of  all  No.  2  rails  shall  be  painted  in  order  to 
distinguish  them. 

STEEL  SPLICE-BARS. 

Adopted  1901. 

PROCESS  OF  MANUFACTURE. 

1.  Steel  for   splice-bars  may  be  made  by  the  Bessemer,  or  open- 
hearth  process. 

CHEMICAL  PROPERTIES. 

2.  Steel  for  splice-bars    shall  conform  to  the  following  limits  in 
chemical  composition: 

Per  Cent. 

Carbon  shall  not  exceed °-I5 

Phosphorus  shall  not  exceed o .  10 

Manganese o .  30-0 . 60 

PHYSICAL  PROPERTIES. 
*.  Splice-bar  steel  shall  conform  to  the  following  physi- 

.  °       ...  Tensile  Tests. 

cal  qualities: 

Tensile  strength,  pounds  per  square  inch 54,ooo  to  64,000 

Yield -point,  pounds  per  square  inch ._....  32,000 

Elongation,  per  cent  in  eight  inches  shall  not  be 

less  than. 25 


482  APPLIED  MECHANICS. 

4.  (a)  A  test  specimen   cut  from  the  head  of  the  splice-bar  shall 
bend  180°  flat  on  itself  without  fracture  on  the  outside  of 

Bending 

Tests.  the  bent  portion. 

(b)  If  preferred,  the  bending  tests  may  be  made  on  an  unpunched 
splice-bar,  which,  if  necessary,  shall  be  first  flattened,  and  shall  then  be 
bent  1 80°  flat  on  itself  without  fracture  on  the  outside  of  the  bent  por- 
tion. 

TEST  PIECES  AND  METHODS  OF  TESTING. 

Test  Sped-  5-  A  test  specimen  of  eight  inch  (8")  gauged  length,  cut 

TeensifierTests.   fr°m  the  head  of  the  splice-bar,  shall  be  used  to  determine 
the  physical  properties  specified  in  paragraph  No.  3. 

6.  One  tensile-test  specimen  shall  be  taken  from  the  rolled  splice- 

bars  of  each  blow  or  melt ;  but  in  case  this  develops  flaws, 
Number  of  or  breaks  outside  of  the  middle  third  of  its  gauged  length, 
Tensile  ests.  ^  mav  ^  discarded,  and  another  test  specimen  substituted 

therefor. 

7.  One  test  specimen  cut  from  the  head  of  the  splice-bar  shall  be 

taken  from  a  rolled  bar  of  each  blow  or  melt,  or,  if  preferred, 
nfen  foi^1"  the  bending  test  may  be  made  on  an  unpunched  splice-bar 
Bending.  which,  if  necessary,  shall  be  flattened  before  testing.  The 
bending  test  may  be  made  by  pressure  or  by  blows. 

8.  For  the  purposes  of  this  specification  the  yield-point  shall  be  de- 

termined by  the  careful  observation  of  the  drop  of  the  beam 
Yield-point.  or  nalt  jn  the  gauge  of  tne  testing  machine. 

9.  In  order  to  determine  if  the  material  conforms  to  the 
Sample  for       chemical  limitations  prescribed  in  paragraph  No.  2  herein, 
Analysis!        analysis  shall  be  made  of  drillings  taken  from  a  small  test 
ingot. 

FINISH. 

10.  All  splice-bars  shall  be  smoothly  rolled  and  true  to  templet. 
The  bars  shall  be  sheared  accurately  to  length  and  free  from  fins  and 
cracks,  and  shall  perfectly  fit  the  rails  for  which  they  are  intended.  The 
punching  and  notching  shall  accurately  conform  in  every  respect  to  the 
drawing  and  dimensions  furnished.  A  variation  in  weight  of  more 
than  2  J  per  cent  from  that  specified  will  be  sufficient  cause  for  rejection. 


STRENGTH   OF  STEEL.  483 

BRANDING. 

11.  The  name  of  the  maker  and  the  year  of  manufacture  shall  be 
rolled  in  raised  letters  on  the  side  of  the  splice-bar. 

INSPECTION. 

12.  The  inspector,  representing  the  purchaser,  shall  have  all  reason- 
able facilities  afforded  to  him  by  the  manufacturer,  to  satisfy  him  that 
the  finished  material  is  furnished  in  accordance  with  these  specifications. 
All  tests  and  inspections  shall  be  made  at  the  place  of  manufacture, 
prior  to  shipment. 

§  226.  Strength  of  Steel. — The  literature  upon  steel  is 
exceedingly  voluminous,  and  many  books  and  articles  written 
upon  the  metallurgy  of  steel,  such  as  "Metallurgy  of  Steel,"  by 
Henry  M.  Howe,  and  "The  Manufacture  and  Properties  of  Iron 
and  Steel,"  by  H.  H.  Campbell,  contain  a  great  many  tests,  which 
have,  as  a  rule,  to  do  with  its  properties  and  the  effects  of  different 
compositions  and  treatments.  They  do  not  often  contain,  how- 
ever, tests  upon  full-size  pieces,  such  as  columns  for  bridges  or 
buildings,  beams,  large  riveted  joints,  full-size  parts  of  machinery, 
etc.  The  greater  part  of  this  latter  class  of  tests  are  to  be  found  in 
the  reports  of  the  various  testing  laboratories,  such  as  those  "of 
the  laboratories  at  Munich,  at  Berlin,  and  at  Zurich  in  Europe, 
and  the  Watertown  Arsenal  reports  and  the  Technology  Quarterly 
in  America;  and  also  in  various  articles  in  the  Proceedings  of 
the  various  Engineering  Societies  in  Europe  and  America.  A 
number  of  these  have  already  been  mentioned  among  the  refer- 
ences to  tests  of  wrought-iron,  and  the  greater  part  of  them  contain 
also  experiments  on  steel. 

References  to  such  full-size  tests  of  steel  as  are  quoted  here 
will  be  given  in  connection  with  the  tests  themselves. 

A  detailed  study  of  the  effect  of  the  different  ingredients 
and  combinations  of  ingredients,  upon  the  strength,  elasticity, 
and  ductility  of  steel,  is  a  very  complicated  matter;  it  belongs 
to  the  study  of  Metallurgy  and  is  beyond  the  scope  of  this 
work.  Nevertheless,  the  engineer  needs,  of  course,  some  general 


484  APPLIED   MECHANICS. 


knowledge  of  these  matters,  and  especially  of  the  effect,  within 
certain  limits,  of  different  percentages  of  carbon. 

This  subject  has  been  dealt  with  by  Mr.  Wm.  R.  Webster 
in  the  Trans.  Am.  Inst.  Mining  Engineers,  of  October,  1892, 
August,  1893,  and  October,  1898,  and  in  the  Journal  of  the  Iron 
and  Steel  Institute,  No.  i,  1894;  also  by  Mr.  A.  C.  Cunningham 
in  the  Trans.  Am.  Soc.  Civil  Engineers  of  December,  1897;  and 
by  Mr.  H.  H.  Campbell,  in  his  book,  "  Metallurgy  of  Iron  and 
Steel."  Of  course  none  of  them  claims  anything  more  than 
approximation  for  their  various  rules  and  formulae,  and  then  only 
in  the  case  of  what  they  call  normal  steel,  i.e.,  such  steel  as  is 
most  frequently  manufactured  by  the  mills. 

Mr.  Webster  made  an  investigation  of  the  effects  of  carbon, 
phosphorus,  manganese,  and  sulp'hur  upon  the  tensile  strength 
of  the  steel.  He  gives  a  set  of  tables  from  which  to  determine, 
approximately,  the  tensile  strength  of  normal  steel,  of  a  given 
chemical  composition.  His  investigations  were  principally  made 
upon  basic  Bessemer,  and  basic  open-hearth  steel. 

Mr.  Campbell  gives  a  formula  for  the  tensile  strength  of  acid, 
and  another  for  the  tensile  strength  of  basic  steel,  and  states  that 
they  represent  the  facts  with  a  good  degree  of  accuracy.  His 
formulae  are  as  follows  : 

For  acid  steel, 

38600  +  1  2iC  +  89?  +  R  =  ultimate  strength; 
For  basic  steel, 

=  ultimate  strength; 


where  C  indicates  carbon,  P  phosphorus,  and  Mn  manganese, 
in  units  of  o.ooi  per  cent,  and  R  depends  upon  the  finishing 
temperature,  and  may  be  plus  or  minus. 

Mr  Cunningham  gives  the  following  rule:  To  find  the  approx- 
imate tensile  strength  of  structural  steel;  to  a  base  of  40000  add 
1000  pounds  for  every  o.oi  per  cent  of  carbon,  and  1000  pounds  for 


STRENGTH  OF  STEEL. 


485 


every  o.oi  per  cent  of  phosphorus,  neglecting  all  other  elements 
in  normal  steel. 

In  this  connection  a  set  of  tests  will  be  quoted  which  were 
made  on  the  government  testing-machine  at  Watertown  Arsenal, 
upon  specimens  of  steel  containing  different  percentages  of  carbon, 
the  tests  themselves  forming  a  portion  of  a  series  denominated 
in  the  government  report  as  the  "Temperature  Series."  The 
account  of  the  tests  to  be  quoted  is  to  be  found  in  their  report 
for  1887. 

Ten  grades  of  open-hearth  steel  are  here  represented,  in  which 
the  carbon  ranges  from  0.09  to  0.97  per  cent,  varying  by  tenths 
of  a  per  cent  as  nearly  as  was  practicable  to  obtain  the  steel. 

The  other  elements  do  not  follow  any  regular  succession. 

TENSILE   TESTS   OF    STEEL    BARS— TEMPERATURE    SERIES. 
Tests  at  Atmospheric   Temperature. 


c 
i) 

a 

I 

$ 

Is 

a 

ss 

3.S 

M 

§n 

1 

1° 
d 

Carbon,  Per  Cent 

Manganese,  Per  C 

Silicon,  Per  Cent. 

Diameter,  Inches. 

Sectional  Area,  Sq 

i. 

Length  of  Rest, 
Months. 

|a 
ojf 

li 

fe 

C/3  . 

&~4 
|W 

Elongation  in  30 
Per  Cent. 

Contraction  of  A 
at  Fracture, 
Cent. 

Mechanical  Work 
Elastic  Limit, 
Inch-Lbs. 

Mechanical  Work 
Tensile  Streng 
in  Inch-Lbs. 

Pounds  per  Sq.  In. 
Ruptured  Sectio 

753 

0.09 

O.II 

1.009 

0.80 

21000 

3 

30000 

52475 

23-6 

63-5 

15-85 

9808.36 

106434 

754 

0.20 

0.45 

1.009 

0.80 

25000 

3 

395oo 

68375 

21.2 

49.1 

26.40 

10651.90 

"3704 

755 

0.31 

0.57 

0.798 

0.50 

25000 

6 

46500 

80600 

18.0 

43-5 

37-27 

10660.77 

126640 

7560.37 

0.70 

0.798 

0.50 

25000 

6 

5OOOO 

85160 

17-5 

45-3 

42.50 

10935-48 

134600 

| 

0.58 

0.02 

0.798 

0.50 

30000 

6 

58000 

98760 

14.9 

41.6 

58.00 

11380.62 

152380 

758 

0-57 

o-93 

0.07 

0.798 

0.50 

30000 

6 

55000 

117440 

10.  I 

14.0 

52-43 

11169.34 

134880 

759 

0.71 

0.58 

0.08 

0-757 

0.45 

35000 

12 

57000 

116000 

8.8 

26.2 

56.53 

9231.21 

151510 

760 

0.81 

0.56 

0.17 

0.798 

0.50 

40000 

12 

70000 

149600 

5-o 

5-4 

84-35 

7872.20 

158140 

761 

0.89 

o-57 

0.19 

0-757 

0-45 

45000 

12 

75000 

141290 

4-3 

4-4 

95-00 

6418.53  147860 

762 

o-97 

0.80 

0.28 

0-757 

0-45 

50000 

12 

79000 

'52550 

4-3 

5-8 

108.62 

7550-23  161910 

486 


APPLIED   MECHANICS. 


The  following  tables  include  sets  of  miscellaneous  tests  of 
various  kinds  of  steel. 


Bessemer  Steel. 

Open-hearth  Steel. 

Q  u- 

'  W 

|   ^ 

i. 

|  ,1 

O  ^v 

S!§ 

5  . 
c« 

S  fc 

3   Q.^, 

t* 

la? 

ojj 
w"G 

If 

1-sjjS 

'S'-  in  C' 

3  §  o 

•si 

0  % 

S-6  „,.= 

^  *^    C 

Ps 

2  '^ 

*j  B 

u  s 

!-s§sT 

$3* 

|*t 

•o  — 
0  W 

!l 

l^esr 

J^sr 

Is 

.7426 

70983 

40397 

54-7 

29139000 

.7600 

64169 

47395 

56.7 

29392600 

.7481 

57700 

33000 

53-5 

28885000 

.7500 

63083 

44137 

64.0 

30179000 

•7463 

58408 

28575 

58-9 

32799000 

.7600 

64477 

47394 

60.  i 

30780000 

.7285 

62761 

34787 

65-2 

32135000 

.7700 

62449 

46171 

63.5 

30481000 

.7476 

50505 

19364 

72-5 

29479000 

.7700 

62556 

46171 

64.3 

29073000 

•7442 

51230 

r9550 

69-7 

30653000 

.7700 

62857 

46171 

59-5 

29073000 

•75°° 

51110 

21503 

71  '5 

28457000 

.7600 

643  '5 

45189 

64.1 

29843000 

•75°° 

51518 

21503 

50.1 

27665000 

.7650 

63527 

44600 

64.6 

29008000 

.7300 

73865 

46584 

56.8 

29600000 

.7600 

64830 

42984 

61.8 

28527000 

.7500 
.7400 
.7400 

50294 
97655 
87086 

26029 
54641 
47666 

27.0 
44.8 
46.8 

18055000 
30539000 
30090000 

•755° 
.7600 

•7575 

65020 
65140 
65240 

45790 
45*90 
43270 

57-9 
64.9 
62.3 

29338000 
31288000 

30040000 

.7600 

87508 
65235 

50673 
49598 

48.0 
62.5 

30057000 
30310000 

•7575 
.7600 

65125 
64500 

45487 
40780 

59  9 
61.9 

29912000 
30060000 

•7350 

87014 

50673 

45-o 

30058000 

•7550 

65089 

41320 

61.3 

291340:0 

.7400 

87356 

49991 

38.6 

30090000 

.87 

43300 

21900 

75-6 

28500000 

•75°° 
.7420 

86720 

48665 
49650 

46.2 
47-2 

28868000 
29887000 

•73 
•72 

44900 
46300 

21500 

22IOO 

75-7 
73-6 

29900000 
29900000 

.7691 

60465 

35526 

61.5 

29149000 

.60 

46800 

24800 

73  '3 

30800000 

•7730 

66077 

35J59 

61.5 

30244000 

.60 

46700 

24800 

75-0 

30000000 

.7690 

66745 

35526 

62.9 

30075000 

.7690 

66445 

39832 

62.5 

30560000 

.7690 

66142 

35526 

60.7 

27864000 

.7680 

66530 

356i8 

60.9 

29225000 

.7690 

67068 

61.5 

30075000 

Machine-steel. 

Boiler-plate. 

<u 
ffjj 

•Si 

W 

i_ 

y  *j  .  c 

C   «-! 

_o  rt-  c 
3  <3  u 

"o'S* 
II 

Section. 

.  •  C 

"2  £>'" 

0i.!U 

!<  £ 

*o  £ 

1-S 

I5 

|3dSf 

Jsdsr 

1»£ 

|s 

°esr 

|;3d5f 

|« 

.7608 

91795 

62693 

53-3 

29316000 

•379   i-458 

5945° 

31670 

47-3 

20459000 

.7629 

96256 

66723 

44-2 

29391000 

.384   1-48 

58770 

39590 

56.3 

30270000 

•  7633 

96767 

65561 

44.1 

29586000 

.365   1.65 

32380 

45-6 

29305000 

•  7520 

92087 

59665 

40-4 

30482000 

.369   1.49 

61657 

37284 

30135000 

•7593 

92091 

62940 

30848000 

.398   1.511 

5437° 

30760 

58.0 

28608000 

•7598 

92191 

58445 

50-4 

28968000 

.376   1.496 

32889 

57-1 

28511000 

•7625 

86941 

62413 

48.7 

28340000 

.4095x1.3647 

5493° 

33"° 

29826000 

.7623 

9575° 

62445 

46.2 

30604000 

•375  Xi.  494 

55173 

29451 

58.5 

28849000 

.7620 
.7620 

96045 
91220 

62495 
62495 

44.6 
51-6 

28802000 
32706000 

.3737x1.4974 
.475  x  1.0295 

54220 
47954 

31270 

g; 

27490000 

.7560 

96684 

63490 

42.8 

29400000 

•  4702X  1.0064 

51035 

67.5 



.7634 

96567 

66638 

43  -3 

30518000 

.4292x1.0235 

5556o 



64.9 



.7609 

92804 

60755 

51-3 

28884000 

.4258x1.0123 

54984 



67.7 

•7597 

86678 

58460 

50-2 

29867000 

.4225  x  1.0025 

60680 



56.3 

.7580 

96119 

54292 

41.8 

29818000 

.4125  x  1.0025 

61500 



55-3 

.  7600 

86741 

58415 

45-7 

28738000 

•  40X  1.02 

5719° 

26552 

58.5 

25800000 

•75T3 

99635 

62032 

27.4 

21813000 

.  50  x  i  .02 

60352 

28431 

58.8 

36199000 

•7613 

106980 

60413 

48.4 

29291000 

.49  X  1.02 

63825 

31010 

52.0 

25273000 

•759° 

96142 

54149 

45-6 

26643000 

.49  X  1.02 

60024 

29012 

58.8 

26677000 

.7699 

94513 

5907  1 

46.6 

28148000 

.50  X  1.02 

59803 

29012 

50-5 

30012000 

.7622 

9H35 

64654 

54-2 

27164000 

.  49  x  i  .  02 

61024 

29012 

58.8 

30012000 

.7613 

86775 

60413 

47-8 

29291000 

•  5GX  1.02 

60393 

29412 

60.2 

29412000 

•  7567 

9.6249 

61150 

45-2 

28776000 

•  49X  1.02 

63625 

30012 

46.1 

31866000 

•7579 

95303 

67606 

44.9 

30005000 

•  39X  1.28 

50480 

26041 

62.5 

32051000 

•754° 

54870 

40.8 

29416000 

.38x  1.27 

53543 

29009 

59-8 

26168000 

•7554 

84990 

45752 

55-5 

29751000 

.41x1.27 

58144 

27846 

50.1 

35455000 

TENSILE   STRENGTH  OF  STEEL. 


487 


BESSEMER  STEEL  WIRE. 

Diameter 
of  Cross- 
section. 
(Inches.) 

Elastic 
Limit. 
(Lbs.  per 
sq.  in.) 

Maximum 
Load. 
(Pounds.) 

Maximum 
Load. 
(Lbs.  per 
sq.  in.) 

Reduction 
of  Area. 
(Per  cent.) 

Modulus  of 
Elasticity. 
(Lbs,  per 
sq.  in. 

.  1290 

1013 

77500 

64.4 

30OCOOOO 

.1280 

66100 

1021 

79400 

55-9 

30400000 

.1288 

68300 

1OIO 

77500 

61.4 

30900000 

.I2yr 

67200 

970 

74100 

63-5 

30000000 

.1283 

66500 

996 

77000 

57-1 

28500000 

.1283 

69600 

IO2I 

79000 

57  -1 

30000000 

.1289 

69700 

I005 

77000 

60.5 

3  i  200000 

.1281 

71400 

I°43 

80900 

63-9 

29200000 

.128, 

65800 

1004 

77700 

62.6 

30700000 

.1286 

68500 

IOOO 

77000 

63.2 

31000000 

BESSEMER  SPRING-STEEL  WIRE. 

.0911 

76000 

9^0 

146000 

34-6 

24500000 

.0910 

69900 

974 

149000 

51  .6 

25900000 

.0905 

79600 

969 

150000 

42.0 

23000000 

.0911 

72900 

95° 

146000 

37-5 

24200000 

.0905 

93i 

143000 

39-6 

25400000 

TESTS  OF  STEEL  EYE-BARS. 

Tests  of  Steel  Eye-bars  made  on  the  Government  Machine. 
— In  the  Tests  of  Metals  at  Watertown  Arsenal  for  1883  is 
the  record  of  the  tests  of  six  eye-bars  of  steel,  presented  by 
the  president  of  the  Keystone  Bridge  Company. 

The  following  is  an  extract  from  the  report  in  regard  to 
these  eye-bars : 

"  The  eye-bars  were  made  of  Pernot  open-hearth  steel,  fur- 
nished by  the  Cambria  Iron  Company  of  Johnstown,  Penn. 

"The  furnace  charges,  about  15  tons  each  of  cast-iron, 
magnetic  ore,  spiegeleisen,  and  rail-ends,  preheated  in  an  aux- 
iliary furnace,  required  six  and  one-half  hours  for  conversion. 

"  All  these  bars  were  rolled  from  the  same  ingot. 

"  Samples  were  tested  at  the  steel-works  taken  from  a  test 
ingot  about  one  inch  square,  from  which  were  rolled  |-inch 
round  specimens. 


488 


APPLIED    MECHANICS. 


"The  annealed  specimen  was  buried  in  hot  ashes  while  still 
red-hot,  and  allowed  to  cool  with  them. 

"  The  following  results  were  obtained  by  tensile  tests  :  — 


Elastic 
Limit,  in 
Ibs.,  per 
Sq.  In. 

Ultimate 
Strength,  in 
Ibs.,  per 
Sq.  In. 

Contrac- 
tion of 
Area. 

Modulus 
of 
Elasticity. 

Carbon. 

f-inch  round  rolled  bar  . 

48040 

73I5° 

%• 

45-7 

28210000 

%- 

0.27 

f-inch  round  rolled  and 

annealed  bar  .... 

422IO 

69470 

54-2 

292IOOOO 

0.27 

"  The  billets  measured  7  inches  by  8  inches,  and  were 
bloomed  down  from  14-inch  square  ingot. 

"  They  were  rolled  down  to  bar-section  in  grooved  rolls  at 
the  Union  Iron  Mills,  Pittsburgh. 

"  The  reduction  in  the  roughing-rolls  was  from  7  inches  by 
8  inches  to  6J  inches  by  4  inches ;  and  in  the  finishing-rolls,  to 
6^  inches  by  I  inch. 

"The  eye-bar  heads  were  made  by  the  Keystone  Bridge 
Company,  Pittsburgh,  by  upsetting  and  hammering,  proceeding 
as  follows  :  — 

"The  bar  is  heated  bright  red  for  a  length  of  (approxi- 
mately) 27  inches,  and  upset  in  a  hydraulic  machine ;  after 
which  the  bar  is  reheated,  and  drawn  down  to  the  required 
thickness,  and  given  its  proper  form  in  a  hammer-die. 

"The  bars  are  next  annealed,  which  is  done  in  a  gas-furnace 
longer  than  the  bars.  They  are  placed  on  edge  on  a  car  in  the 
annealing-furnace,  separated  one  from  another  to  allow  free 
circulation  of  the  heated  gases.  They  are  heated  to  a  red  heat, 
when  the  fires  are  drawn,  and  the  furnace  allowed  to  cool. 
Three  or  four  days,  according  to  conditions,  are  required  before 
the  bars  are  withdrawn. 


TENSILE  STRENGTH  OF  STEEL.  489 

"  The  pin-holes  are  then  bored. 

"The  analyses  of  the  heads  before  annealing  were:  — 

"  Carbon,  by  color 0.270  per  cent 

Silicon 0.036        " 

Sulphur °-°75        " 

Phosphorus 0.090       " 

Manganese 0.380       " 

Copper Trace. 

"  The  bars  were  tested  in  a  horizontal  position,  secured  at 
the  ends,  which  were  vertical. 

"  To  prevent  sagging  of  the  stem,  a  counterweight  was  used 
at  the  middle  of  the  bar. 

"  Before  placing  in  the  testing-machine,  the  stem  from  neck 
to  neck  was  laid  off  into  lo-inch  sections,  to  determine  the 
uniformity  of  the  stretch  after  the  bar  had  been  fractured. 

"A  number  of  intermediate  lo-inch  sections  were  used  as 
the  gauged  length,  obtaining  micrometer  measurements  of 
elongation,  and  the  elastic  limit  for  that  part  of  the  stem  -which 
was  not  acted  upon  during  the  formation  of  the  heads.  Elon- 
gations were  also  measured  from  centre  to  centre  of  pins,  taken 
with  an  ordinary  graduated  steel  scale. 

"  The  moduli  of  elasticity  were  computed  from  elongations 
taken  between  loads  of  10000  and  30000  Ibs.  per  square  inch, 
deducting  the  permanent  sets. 

"The  behavior  of  bars  Nos.  4582  and  4583,  after  having 
been  strained  beyond  the  elastic  limit,  is  shown  by  elongations 
of  the  gauged  length  measured  after  loads  of  40000  and  50000 
Ibs.  per  square  inch  had  been  applied ;  and  with  bar  No.  4583, 
after  its  first  fracture  under  64000  Ibs.  per  square  inch,  a  rest 
of  five  days  intervening  between  the  time  of  fracture  and  the 
time  of  measuring  the  elongations. 

"Considering  the  behavior  between  loads  of  roooo  and 
30000  Ibs.  per  square  inch,  we  observe  the  elongations  for  the 


49°  APPLIED   MECHANICS. 

primitive  readings  are  nearly  in  exact  proportion  to  the  incre- 
ments of  load. 

"  Loads  were  increased  to  40000  Ibs.  per  square  inch,  passing 
the  elastic  limit  at  about  37000  Ibs.  per  square  inch  ;  the  respec- 
tive permanent  stretch  of  the  bars  being  1.31  and  1.26  per  cent. 

"  Elongations  were  then  immediately  redetermined,  which 
show  a  reduction  in  the  modulus  of  elasticity,  as  we  advanced 
with  each  increment,  of  5000  Ibs.  per  square  inch. 

"  Corresponding  measurements  after  the  bars  had  been 
loaded  with  50000  Ibs.  per  square  inch  reach  the  same  kind  of 
results. 

"The  first  fracture  of  bar  No.  4583,  under  64000  Ibs.  per 
square  inch,  occurred  at  the  neck,  leaving  sufficient  length  to 
grasp  in  the  hydraulic  jaws  of  the  testing-machine,  and  con- 
tinue observations  on  the  original  gauged  length.  This  was 
done  after  the  fractured  bar  had  rested  five  days. 

"  The  elongations  now  show  the  modulus  of  elasticity  con- 
stant or  nearly  so,  the  only  difference  in  measurements  being 
in  the  last  figures,  up  to  50000  Ibs.  The  readings  were  then 
immediately  repeated,  and  the  same  uniformity  of  elongations 
obtained. 

"  An  illustration  of  the  serious  influence  of  defective  metal 
in  the  heads  is  found  in  the  first  fracture  of  bar  No.  4583. 

"  There  was  about  27  per  cent  excess  of  metal  along  the 
line  of  fracture  over  the  section  of  the  stem." 


TENSILE  STRENGTH  OF  STEEL. 


49 I 


z 

s  * 

£ 

2 

W*              W^ 

^*             ft 

S  » 

!?*(. 

8.    8. 

£ 

w 

g     S 

3-    fr 

*                 f* 

r* 

J* 

1          1 

r*     ^ 

=r  5  ,S 

W          N 

<s? 

•3 

f    S 

8      S, 

8     8 

8 

§ 

•        8- 

§-  r 

Gauged  Length,  in  inches. 

C  1" 

i 

C 

1           1 

ON       00 

Width,  in  inches. 

p    p 

vO       <O 

V|            V| 

p 

p 

1          1 

P         P 

N! 

j<       .w  PT 

W         W 

OJ 

(A) 

w        w 

^w      5V_w 

O>         M 

CO 

£ 

1           1 

oJ     ^ 

o    C  "ft    -;  §    ^ 

0        0 

0 

8 

0        0° 

•  s     j"  -"  s1 

If 

i 

1 

"3      ^ 

0            £ 

<8        o* 

f  ! 

[fvril 

OJ         Ov 

M 

M 

8 

M 
10 

^ 

OJ 

1  1 

In    Gauged            W 
Length.                »  J 

<2        <£ 

M 

M 

1       ^* 

Centre  to  Cen-        S    5- 

in         M 

w 

0 

'        vb 

tre  of  Pins.          '    •? 

•*•        -f" 

% 

S 

00                ON 

W 

Contraction  of  Area,  per 

*•        *" 

w 

M 

*.          <J\ 

^ 

cent. 

NO         ON 

f 

S 

1          I 

it 

W 

ff       3 

f 

s 

1           1 

2     » 

Maximum      Compression 
on  Pin-Holes,   in   Ibs., 

8s    8s 

6 

o 

o    8 

per  Square  Inch. 

W 

ed    w 

s      s 

8 

5 

*         1 
n 

1  1 

*J* 

s      s 

S 

S 

R         5' 

S     5' 

3   I- 

5        5 

a 

8 

&     8 

s     • 

H 

S- 

3      ?. 

« 

£fi 

^^  ^ 

O     Jrt 

^ 

8    Z 

p 

JC 

«< 

1    8     « 

i  Is 

2 

II 

5r 
vj 

1 

w 

ff.    <«     « 
P    I  f 

c      rj 

?  g. 

i 

I  ? 

t> 

i 

1 

S   s 

«      o- 

c 

3, 

i 

^ 

$ 

2,    ? 

55 

31 

o 

3 

W        B 

O 

? 

» 

?        P 

p 

492 


APPLIED   MECHANICS. 


ELONGATIONS    OF   No.  4582    FOR   EACH    INCREMENT   OF   5000  LBS.   PER 

SQUARE    INCH. 


Loads,  in  Ibs., 
per 
Square  Inch. 

Elongations. 

Primitive  Load- 
ing. 

After  Load  of 
40000  Ibs.  per 
Square  Inch. 

After  Load  of 
50000  Ibs.  per 
Square  Inch. 

IOOOO 

_ 

. 

_ 

15000 
20000 
25000 

0.0274 
0.0269 
0.0269 

0.0300 
0.0305 
0.0320 

0.03II 
0.0322 
0.0337 

30000 

0.0269 

0.0330 

0.0341 

ELONGATIONS    OF   No.  4583    FOR  EACH    INCREMENT   OF   5000  LBS.  PER 

SQUARE    INCH. 


Loads,  in 

Elongations. 

Elongations  after  64000  Ibs.  per 
Square  Inch. 

Ibs.,  per 

Square  Inch. 

Primitive  Load- 

After 40000  Ibs. 

After  50000  Ibs. 

First 

Second 

ing. 

per 
Square  Inch. 

per 
Square  Inch. 

Reading. 

Reading. 

IOOOO 

_ 

. 

. 

. 

. 

15000 

0.0272 

0.0291 

0.0302 

0.0311 

0.0310 

2OOOO 

0.0272 

0.0305 

0.0315 

0.0308 

0.0310 

25OOO 

0.0268 

0.0314 

0.0325 

0.0311 

0.0310 

3OOOO 

0.0267 

0.0326 

0.0340 

0.0312 

0.0310 

35000 

- 

- 

- 

0.0311 

- 

40000 

- 

- 

- 

0.0312 

- 

45000 

- 

- 

- 

0.0310 

- 

50000 

" 

~ 

" 

0.0315 

In  the  Tests  of  Metals  for  1886  is  given  the  following  table 
of  tensile  tests  of  steel  eye-bars,  furnished  by  the  Chief  Engineer 
of  the  Statue  of  Liberty. 


TENSILE  STRENGTH  OF  STEEL. 


493 


Dimensions. 

ft 

ft 

Elongation. 

i 

£  . 

*l- 

Fractxire. 

o  , 

f< 

. 

+3  o 

Q,    rj      O 

|| 

V 

M 

*o 

c 

J~ 

cS(§^ 

-S 

bo  §"0 

F 

Width. 

Thickness 

Elastic 
Square 

Tensile  S1 
Square 

it 
fj 

Center  to 
of  Pin-1 

Contracti 

Modulus  c 
per  Sqx 

Maximun 
sion  on 
per  Sqx 

Location. 

Appearance. 

Ins. 

Ins. 

Ins. 

Lbs. 

Lbs. 

% 

% 

% 

Lbs. 

Lbs. 

308.00 

5-  16 

I  .02 

34610 

64870 

7-4 

7-3 

31400000 

74173 

308.00 

5-14 

I  .02 

34730 

69330 

10.4 

10.3 

29279000 

84093 

308.00 

5-15 

I  .02 

37330 

70286'  11.7 

ii.  5 

29017000 

80043 

•308.  10 

5-14 

I  .02 

35000 

70229 

ii.  6 

ii-4 

13-4 

79826 

Stem 

Granular,  radi- 

ating from  a 

button    of 

hard  metal. 

308.00 

5-13 

I  .02 

35950 

71680 

ii.  8 

ii.  5 

81323 

307.95 

5-iS 

I  .02 

35000 

70895 

12.  I 

ii.  8 

30162000 

80737 

The  gauged  length  of  the  bars  was  260  inches.  The  moduli 
of  elasticity  computed  between  25000  and  30000  pounds  per 
square  inch. 

In  connexion  with  the  work  upon  the  bridge  over  the  Missis- 
sippi at  Memphis,  Mr.  Geo.  S.  Morison,  the  Chief  Engineer, 
had  56  full-size  stee  eye-bars  tested.  The  results  are  given  in 
his  Report,  dated  March,  1894,  and  furnish  valuable  information 
regarding  the  behavior  of  the  steel,  and  the  design,  and  con- 
struction of  the  bars.  Only  the  following  table  (see  page  494) 
will  be  given  here,  containing  a  portion  of  the  results  of  the  tests 
upon  31  of  the  bars,  all  made  of  basic  open-hearth  steel,  and 
all  of  which  broke  in  the  body. 

This  table  will  aid  the  reader  in  comparing  the  tensile  strength 
and  the  limit  of  elasticity  of  full-size  steel  eye-bars,  with  those 
obtained  from  the  tests  of  small  samples  of  the  steel. 

In  Engineering  News  of  Feb.  2,  1905,  'is  an  article  containing 
a  comparison  of  full-size  and  specimen  tests  of  eleven  steel  eye- 
bars,  made  at  the  Phoenix  Iron  Co.  Each  of  these  bars  was  15 
inches  wide;  two  of  them  were  ij  inches  thick;  one  was  i& 
inches  thick,  six  were  2  inches  thick,  and  two  were  2&  inches 
thick.  The  specimen  tests  gave  tensile  strengths  varying  from 
60310  to  67000  pounds  per  square  inch,  and  limits  of  elasticity 
varying  from  31550  to  41760  pounds  per  square  inch. 


494 


APPLIED    MECHANICS. 


FULL-SIZE  EYE-BARS. 


SAMPLE  BARS  FROM  SAME  MELT. 


u 

•u 

J. 

i 

Jl 

Ii 

£     . 

jd 

a 
44 

£o 

•Js^ 

ff 

£ 

c  O 

W    & 

*& 

P 

'1 

JT 

3 

Ins. 

Ins. 

Ins. 

Lbs. 

Lbs. 

10.07 

1.50 

160.63 

35ICO 

67490 

9-95 

1-73 

358.93 

3768o 

70160 

9.98 

i-75 

361-23 

39700 

65500 

10.05 

1.50 

162.38 

33140 

65060 

6.08 

291.26 

29690 

56700 

10.07 

iley 

287.37 

32860 

65600 

9.92 

284.28 

31110 

61060 

9-94 

0-99 

287.88 

3399° 

63220 

10.05 

2.20 

222.88 

2933° 

63100 

10.  12 

1.86 

464.03 

31970 

53860 

7-12 

1.17 

314.04 

30270 

51500 

IO.07 

2.20 

338.73 

28080 

55160 

10.03 

I.8l 

25L58 

29670 

62140 

9-97 

i-37 

250.28 

32700 

65400 

7.02 

385.73 

28980 

52010 

7.01 

i!26 

385.78 

28410 

54740 

9-99 

1.62 

249.98 

30500 

58870 

9.96 

2.05 

341.28 

3336o 

7355° 

10.13 

1.30 

249.48 

32520 

60710 

9.98 

1.81 

284.82 

28000 

58720 

10.  15 

1-83 

221  .98 

32290 

62270 

10.04 

o-99 

361.68 

29970 

58680 

7.01 

1.27 

258.68 

28640 

56830 

7.98 

i  .20 

254.63 

31930 

63870 

8.03 

2.32 

338.58 

32840 

62400 

7.00 

1.18 

258.68 

27870 

53520 

9.09 

1.25 

206.58 

3259° 

574io 

8.  ii 

1.79 

279.98 

28940 

58010 

7.00 

i  .00 

289.23 

31380 

59850 

t 

•3d 

i-5  c 

I* 

£-"+J 

e~ 

P  •*-* 

"3 
| 

Is 

So* 

If 

fs 

1 

W 

Js 

w 

1 

O, 

Sq.  In. 

Lbs. 

Lbs. 

•9500 

27-5 

41580 

73050 

.027 

.9918 

24.4 

42650 

75620 

•015 

.9520 

28.8 

40280 

70280 

.062 

.9500 

27-5 

41580 

73050 

.027 

•9756 

28.1 

40490 

69700 

.026 

1.1590 

20.  o 

43750 

75000 

.021 

1.0140 

28.8 

42210 

69730 

.046 

.9868 

28.1 

40230 

69720 

.025 

•9635 

28.8 

38090 

71300 

.017 

I  .O2OI 

27.0 

40200 

71860 

.017 

I  .0180 

28.8 

33400 

57170 

.014 

.  I22O 

24.2 

38320 

70220 

.023 

.O20O 

26.3 

40200 

71080 

.028 

.0670 

25.0 

3936o 

69360 

.041 

.  I70O 

31-3 

34190 

58460 

•039 

.0170 

28.1 

41400 

67840 

.OIO 

•9338 

25.0 

40910 

70360 

.014 

.9700 

25-5 

40410 

69900 

.063 

•95°4 

27.0 

40400 

70490 

.023 

•5557 

29-5 

40000 

66800 

.008 

.9746 

21.3 

40530 

72240 

.056 

.1720 

27.0 

40610 

70480 

.O6O 

.0200 

28.! 

4079° 

68730 

.030 

.0100 

21.9 

40900 

69800 

.024 

.0620 

23.1 

41710 

71000 

.066 

.0560 

32480 

58050 

.027 

•9734 

28^7 

38110 

60920 

.014 

.114 

23.0 

40480 

66880 

.030 

.020 

28.1 

40790 

68730 

.030 

The  decrease  of  tensile  strength  in  the  full-size  eye-bars 
varied  from  6.3  per  cent  to  11.9  per  cent,  while  the  decrease  in 
elastic  limit  varied  from  8.3  per  cent  to  17  per  cent. 

STEEL   COLUMNS. 

In  the  Trans.  Am.  Soc.  C.  E.,  of  June,  1889,  will  be  found  a 
paper  by  Mr.  J.  G.  Dagron,  giving  an  account  of  a  set  of  tests  of 
eight  full-size  Bessemer-steel  bridge  columns,  made  for  the  Sus- 


STEEL    COLUMNS. 


495 


quehanna  River  Bridge  of  the  Baltimore  and  Ohio  R.R.  The  steel 
varied  in  tensile  strength  from  83680  to  84440  pounds  per  square 
inch,  in  elastic  limit  from  51190  to  53890  pounds  per  square  inch, 
in  elongation  in  8  inches  from  18.75  Per  cent  to  2O-75  Per  cent, 
and  in  contraction  of  area  from  30.55  per  cent  to  39.7  per  cent. 
The  columns  were  made  by  the  Keystone  Bridge  Company  and 
tested  in  their  hydraulic  press,  with  the  columns  in  a  horizontal 
position,  and  with  the  pins  horizontal. 

The  results  obtained  are  given  by  the  accompanying  table : 


No.  of 
Column. 

Depth. 
Inches. 

Sectional 
Area. 
Sq.  Ins. 

Length 
Center  to 
Center 
Pin-holes. 

Ratio  of 
Length  to 
Radius  of 
Gyration. 

Square  of 
Radius  of 
Gyration. 

Ultimate 
Strength, 
in  Lbs. 
per 
Sq.  In. 

Modulus  of 
Elasticity. 
Lbs. 
per  Sq.  In. 

I 

8 

8.24 

i6'o' 

42.05 

20.86 

41020 

27705000 

2 

8 

8.24 

i6'o' 

42.05 

20.86 

41650 

27705000 

3 

8 

8.24 

20'  0' 

52  -564 

20.86 

39440 

26113000 

4 

8 

8.24 

20' 

52.564 

20.86 

41050 

25816000 

5 

8 

8.24 

24'0' 

63.075 

20.86 

40230 

29504000 

6 

8 

8.24 

24'0' 

63-075 

20.86 

40070 

28398000 

7 

9 

13-23 

25'7-T 

5s  -795 

27-34 

35570 

26557000 

8 

9 

13-23 

25-7-i" 

58.795 

27.34 

38810 

29478000 

The  columns  failed  as  follows : 


i. 


No. 


No. 


No.  3. 
No.  4 

No.  5 


Failed  by  bending  downwards  at  rivet  in  latticing,  i  foot 

loj  inches  from  the  center,  buckling  flange  angles  and 

web-plate. 
2.  Failed  by  bending  upwards  at  rivet  in  latticing  at  center, 

buckling    flange    angles    and    web-plate.     One    angle 

was  fractured  at  point  of  buckling,  and  also  at  the  two 

adjacent  rivets  in  latticing 
Failed  by  bending  upwards  between  latticing,  3  feet  from 

center,  buckling  flange  angles  and  web-plate. 
Failed  by  bending  upwards  between  latticing,  4  inches  from. 

center,  buckling  flange  angles  and  web-plate. 
Failed  by  bending  upwards  between  latticing,  9^  inches 

from  center,  buckling  flange  angles  and  web-plate. 


49<5 


APPLIED   MECHANICS. 


No.  6.  Failed    by    bending    upwards    between   latticing,  i    foot 

5!  inches  from  center,  buckling  flange  angles  and  web 

plate. 
No.  7.  Failed  by  bending  upwards  at  rivet  in  latticing,  3  inches 

from  center,  buckling  flange  angles  and  web-plate. 
No.  8.  Failed  by  bending  upwards  at  rivet  in  latticing,   i  foot 

from  center,  buckling  flange  angles  and  web-plate. 

In  every  case,  after  test,  the  rivets  of  each  column  were  found 
by  hammer  test  to  be  perfectly  right. 

The  following  table  gives  the  results  of  a  set  of  tests  by  direct 
compression,  of  eight  connecting-rods  specially  made  for  these 
tests,  by  the  Baldwin  Locomotive  Works,  and  tested  in  the  Labora- 
tory of  Applied  Mechanics  of  the  Mass.  Institute  of  Technology. 


Breaking- 

Area. 

Tensile  Properties  of  the  Steel. 

strength  per  Sq. 

o  *. 

In.  of  the  Rod. 

,3 

oo 

!«3 

J_ 

g 

g 

'§(3 

bo 

£ 

g 

Modulus 

d 
o 

c 

-i< 

P 

"o 

"o 

Id- 
•*w 

ll 

SO" 

J 

'li 

of 
Elasticity 
per 

1 

o 

1 

f<§3 

1 

re 

J"^  % 
ft 

|& 

£    C 

Sq.  In. 

1 

1 

M 

M 

§ 

H 

w 

O 

M 

Ins. 

Sq.In. 

Sq.In. 

Lbs. 

Lbs. 

Pr.Ct. 

Pr.  Ct. 

Lbs. 

.Lbs. 

Lbs. 

A 

89-38 

100.5 

7.19 

7.6o 

57730 

80280 

25-8 

30.9 

28000000 

38700 

36700 

B 

98-38 

109.4 

7.19 

7.78 

4565o 

78830 

20.8 

34.1 

28300000 

40600 

37500 

C 

107.38 

118.5 

6-73 

7.21 

43900 

77840 

20.4 

42.5 

30000000 

39300 

36700 

D 

in-75 

125  .0 

7.27 

7.78 

4.7560 

79270 

22.3 

43-2 

28500000 

36100 

33700 

E 

116.25 

130.0 

7.38 

7.96 

45820 

79250 

30500000 

39300 

36400 

F 

120.63 

134.8 

7.21 

7-55 

49440 

81660 

24.1 

39-9 

28800000 

39300 

37500 

G 

125-13 

139-7 

7.06 

3959° 

79690 

24.4 

45-5 

30300000 

38000 

35000 

H 

134-13 

149.4 

7.28 

7*78 

39470 

78650 

21.0 

28.3 

30800000 

37400 

35°oo 

TRANSVERSE  STRENGTH  OF  STEEL. 


The  following  table  gives  the  results  of  tests  of  a  number  of 
steel  I  beams,  made  in  the  Laboratory  of  Applied  Mechanics 
of  the  Mass.  Institute  of  Technology. 


^SoEVBE^ 

UNIVERSITY 


OF 


TRANSVERSE  S: 


TGTH  OF  STEEL. 


497 


Mo- 

Modulus 

Mo- 

Break- 

dulus of 

of 

No.  of 
Test. 

Depth. 
Inches. 

ment  of 
Inertia. 

Span. 
Feet  and 

ing 
Load. 

Rup- 
ture 

Elasticity 
per 

Remarks. 

Ins. 

Inches. 

Lbs. 

per 

Sq.  In. 

Sq.  In. 
Lbs. 

Lbs. 

290 

7 

38.00 

14'     6" 

10500 

42874 

29030000 

From  Phoenix  Co. 

293 

8 

57-11 

14'     6" 

14200 

44270 

29410000 

t  t            1  1         it 

295 

9 

81.34 

14'     6" 

16700 

40200 

29890000 

n            1  1         it 

337 

6 

24.86 

14'      7" 

8200 

44900 

28170000 

N.  J.  Iron  &  Steel  Co. 

340 

7 

39.63 

12'    II" 

I2OOO 

42100 

27480000 

<  <        if             <  <       it 

343 

8 

5J-67 

14'     7" 

14900 

46400 

29040000 

11                 I  (                           <  C              (I 

63ia 

10 

i34.oo 

14'    o" 

24200 

38500 

28400000 

Carnegie  Steel  Co. 

638 

10 

134.00 

14'     o" 

25100 

395°° 

29300000 

(  (            i  i       t  ( 

674 

IO 

129.00 

14'     o" 

249OO 

41300 

27450000 

1C                        C  C             t  I 

675 

IO 

131  .20 

14'     o" 

25600 

41700 

27850000 

C  (                        t  (              t  I 

In  Heft  IV  of  the  Mitth.  der  Materialprtifungsanstalt  in 
Zurich  are  given  the  following  results  of  tests  of  the  transverse 
strength  of  ten  steel  plate  girders  : 


Depth  of 
Web. 
(Inches.) 

Span. 
(Inches.) 

Modulus  of 
Rupture. 
(Lbs.  per 
sq.  in.) 

Modulus  of 
Elasticity. 
(Lbs.  persq.in.) 

19.76 

177.17 

53325 

29193660 

19.76 

I77.I7 

55316 

27430380 

15-75 

I4L73 

55174 

26662500 

15-75 

I4L73 

55316 

28738620 

19.69 

177.17 

53325 

29193560 

19.69 

177.17 

55316 

27430380 

23.62 

2I2.6O 

57591 

28795500 

23.62 

2I2.6O 

52472 

28155600 

27.56 

248  .  03 

54320 

27529920 

27.56 

248.03 

53041 

28752840 

498  APPLIED   MECHANICS. 


COLD    CRYSTALLIZATION   OF   IRON   AND   STEEL. 

The  question  of  cold  crystallization  of  wrought-iron  and 
steel  is  one  that  has  been  agitated  from  the  earliest  times,  and, 
although  Kirkaldy  tried  to  dispose  of  it  finally  by  offering  evi- 
dence showing  that  it  does  not  exist,  nevertheless  we  find  the 
same  old  question  cropping  out  every  little  while,  and  although 
the  bulk  of  the  evidence  is  admitted  to  be  against  it,  and,  as  it 
seems  to  the  writer,  there  is  no  evidence  in  its  favor,  we  find 
every  now  and  then  some  one  who  thinks  that  certain  observed 
phenomena  can  be  explained  in  no  other  way. 

The  most  usual  phenomenon  which  cold  crystallization  is 
called  upon  to  explain  is  the  crystalline  appearance  of  the 
fracture  of  some  piece  of  wrought-iron  or  steel  that  has  been 
in  service  for  a  long  time,  and  which  has,  as  a  rule,  been  sub- 
jected to  more  or  less  jars  or  shocks.  The  cases  most  fre- 
quently cited  are  those  of  axles  of  some  sort  which  have  been 
broken,  and,  in  the  case  of  which,  the  fracture  has  had  a  crys- 
talline appearance,  and  where  samples  cut  from  the  other  parts 
of  the  axle  and  tested  have  shown  a  fibrous  fracture.  The 
assumption  has  therefore  been  made  that  the  iron  was  origi- 
nally fibrous,  and  that  crystallization  has  been  caused  by  the 
shocks  or  the  jarring  to  which  it  has  been  subjected  in  the 
natural  service  for  which  it  was  intended. 

Kirkaldy  showed  (see  his  sixty-six  conclusions)  that  when 
fibrous  iron  was  broken  suddenly,  or  when  the  form  of  the 
piece  was  such  as  not  to  offer  any  opportunity  for  the  fibres  to 
stretch,  the  fibres  always  broke  off  shorthand  the  fracture  was 
at  right  angles  to  their  length,  and  hence  followed  the  crystal- 
line appearance ;  whereas  if  the  breaking  was  gradual,  and  the 
fibres  had  a  chance  to  stretch,  they  produced  a  fibrous  appear- 
ance :  in  short,  he  claimed  that  the  difference  between  the  crys- 
talline or  the  fibrous  appearance  of  the  fracture  was  only  a 


COLD    CRYSTALLIZATION   OF  IRON  AND    STEEL.         499 

difference  of  appearance,  and  not  a  change  of  internal  structure 
from  fibrous  to  crystalline. 

The  facts  that  Kirkaldy  showed  in  this  regard  are  generally 
acknowledged  to-day,  and  doubtless  answer  by  far  the  greater 
part  of  those  who  claimed  cold  crystallization  at  the  time  that 
he  wrote,  and  also  a  great  many  of  those  who  claim  its  exist- 
ence to-day. 

But  it  is  easy,  if  suitable  means  be  taken,  to  distinguish 
cases  of  crystalline  appearance  of  fracture  from  cases  where 
there  are  actual  crystals  in  the  piece  ;  and  it  is  rather  about 
those  cases  where  the  iron  near  the  fracture  actually  contains 
distinct  crystals  that  what  discussion  there  is  to-day  that  is 
worth  considering  takes  place. 

The  number  of  such  cases  is,  of  course,  small,  but  every 
once  in  a  while  some  one  is  cited,  and  the  claim  is  put  forward 
that  the  iron  was  originally  fibrous,  and  that  these  crystals 
must  therefore  have  been  produced  without  heating  the  iron 
to  a  temperature  where  chemical  change  is  known  to  occur. 

Inasmuch  as  the  one  who  claims  the  existence  of  cold  crys- 
tallization is  announcing  a  theory  which  is  manifestly  opposed 
to  the  well-known  chemical  law  that  crystallization  requires 
freedom  of  molecular  motion,  and  hence  can  only  take  place 
from  solution,  fusion,  or  sublimation,  it  follows  that  the  burden 
of  proof  rests  with  him,  and  before  he  can  substantiate  his 
theory  in  any  single  case  he  must  prove  beyond  the  possibility 
of  doubt,  i°,  that  the  iron  or  steel  was  originally  fibrous,  i.e., 
not  only  that  fibrous  iron  was  used  in  manufacturing  the  pieces, 
but  also  that  it  had  not  been  overheated  during  its  manufac- 
ture, and,  2°,  that  it  has  never  been  overheated  during  its  period 
of  service. 

It  is  because  the  writer  is  not  aware  of  any  case  where  these 
two  circumstances  have  been  proved  to  hold  that  he  says  that 
he  knows  of  no  evidence  for  cold  crystallization.  In  this  con- 
nection it  is  not  worth  while  to  quote  very  much  of  the  exten- 


500  APPLIED    MECHANICS. 

sive  literature  on  the  subject ;  hence  only  a  little  of  the  most 
modern  evidence  will  be  given  here. 

On  page  1007  et  seq.  of  the  report  of  tests  on  the  govern- 
ment testing-machine  at  Watertown  Arsenal  for  1885  is  given 
an  account  of  a  portion  of  a  series  of  tests  upon  wrought-iron 
railway  axles,  and  the  following  is  quoted  from  that  report :  — 

"  This  series  of  axle  tests,  begun  September,  1883,  is  carried 
on  for  the  purpose  of  determining  whether  a  change  in  struc- 
ture takes  place  in  a  metal  originally  ductile  and  fibrous  to  a 
brittle,  granular,  or  crystalline  state,  resulting  from  exposure 
to  such  conditions  as  are  met  with  in  the  ordinary  service  of  a 
railway  axle. 

"  Twelve  axles  were  forged  from  one  lot  of  double-rolled 
muck-bars,  and  in  their  manufacture  were  practically  treated 
alike.  Each  axle  was  made  from  a  pile  composed  of  nine  bars, 
each  6  in.  wide,  f  in.  thick,  and  3  ft.  3  in.  long,  and  was  finished 
in  four  heats,  two  heats  for  each  end. 

"  The  forging  was  done  by  the  Boston  Forge  Company  in 
their  improved  hammer  dies,  which  finish  the  axle  very  nearly 
to  its  final  dimensions. 

"Two  axles  were  taken  for  immediate  test,  to  show  the 
-quality  of  the  finished  metal  before  it  had  performed  any  rail- 
way service,  and  serve  as  standards  to  compare  with  the 
remaining  ten  axles,  to  be  tested  after  they  had  been  in 
use. 

"  The  axles  are  in  use  in  the  tender-trucks  of  express  loco- 
motives of  the  Boston  and  Albany  Railroad.  Mr.  A.  B.  Under- 
hill,  superintendent  of  motive-power,  contributes  the  axles  and 
furnishes  the  record  of  their  mileage." 

The  results  of  some  measurements  of  deflection  are  given 
concerning  one  of  the  axles  in  tender  134,  after  it  had  run 
95000  miles  ;  and  then  follows  :  — 

"  Regarding  the  axle  for  the  time  being  as  cylindrical,  3.96 


COLD    CRYSTALLIZATION  OF  IRON  AND    STEEL.        50 1 


inches  diameter,  the  modulus  of  elasticity  by  computation  will 
be  28541000  pounds. 

"  Applying  this  modulus  to  the  deflections  observed  in  rear 
axle  of  the  rear  trucks  of  tender  No.  150,  the  maximum  fibre 
strain  is  found  to  be  9935  pounds  per  square  inch  when  the 
tender  was  partially  loaded,  and  14900  pounds  per  square  inch 
when  fully  loaded. 

"  Taken  together,  the  tensile  and  compressive  stresses, 
which  are  equal,  amount  to  19870  and  29800  pounds  per  square 
inch  respectively,  as  the  range  of  stresses  over  which  the  metal 
works. 

"  This  definition  of  the  limits  of  stresses  must  be  regarded 
as  approximate.  There  are  influences  which  tend  to  increase 
the  maximum  fibre  strain,  such  as  unevenness  of  the  track,  the 
side  thrust  of  the  wheel-flanges  against  the  rails.  On  the  other 
hand,  the  inertia  of  the  axle,  particularly  under  high  rates  of 
speed,  would  exert  a  restraining  influence  on  the  total  deflec- 
tion. 

"  Nine  tensile  specimens  were  taken  from  each  axle  ;  three 
from  each  end,  including  the  section  of  axle  between  the  box 
and  wheel  bearings,  and  three  from  the  middle  of  its  length. 
They  are  marked  M.B.,  with  the  number  of  the  axle  ;  also  a 
sub-number  and  letter  to  indicate  from  what  part  of  the  axle 
each  was  taken. 

"  The  tensile  test-pieces  showed  fibrous  metal,  and  generally 
free  from  granulation. 

"  The  muck-bar  had  a  higher  elastic  limit  and  lower  tensile 
strength,  and  less  elongation  than  the  axles.  The  moduli  of 
elasticity  of  the  two  are  almost  identical. 

"  Between  loads  of  15000  and  25000  pounds  per  square  inch 
the  muck-bar  had  a  modulus  of  elasticity  of  29400000  pounds, 
the  axles  (average  of  all  specimens)  between  5000  and  20000 
pounds  per  square  inch  was  29367000  pounds.  Individually 


502  APPLIED    MECH AXILS. 

the  axles  showed  the  modulus  of  elasticity  to  be  substantially 
the  same  in  each." 

Two  specimens  were  subjected  to  their  maximum  load  and 
removed  from  the  testing-machine  before  breaking  in  order  to 
see  whether  the  straining  followed  by  rest  will  cause  any 
change. 

"  It  does  not  appear  from  these  tests  that  95000  miles 
run  has  produced  any  effect  on  the  quality  of  the  metal." 

On  page  1619  et  seq.  of  the  Report  for  1886  is  given  an 
account  of  the  tests  made  on  some  more  of  these  axles  which 
had  run  163138  miles,  and  the  following  is  quoted  from  that 
account :  — 

"  Specimens  from  muck-bar  axle  No.  4  after  the  axle  had 
run  163138  miles. 

"  Comparing  these  results  with  earlier  tests  of  this  series,  the 
tensile  strength  of  the  metal  in  this  axle  is  lower,  and  the 
modulus  of  elasticity  less  than  shown  by  the  preceding  axles. 

"  The  variations  in  strength,  elasticity,  and  ductility  are  no 
greater,  however,  than  those  met  in  different  specimens  of  new 
iron  of  nominally  the  same  grade,  and  while  apparently  there 
is  a  deterioration  in  quality,  it  needs  confirmation  of  a  more 
decisive  nature  from  the  remaining  axles  before  attributing 
this  result  to  the  influence  of  the  work  done  in  service." 

Another  set  of  tests  made  at  Watertown  Arsenal  is  to  be 
found  on  page  1044  et  seq.  of  the  Report  for  1885.  There  were 
tested  - 

i°.  Two  side-rods  of  a  passenger  locomotive  which  had  been 
in  service  about  twelve  years. 

2°.  One  side-rod  of  a  passenger  engine  which  had  been  run 
twenty-eight  years  and  eight  months. 

3°.  One  main-rod  which  had  been  run  thirty-two  years  and 
eight  months  in  freight  and  five  years  in  passenger  service. 

In  none  of  these  tests  were  there  any  evidences  of  crystal- 
lization, as  the  metal  was  in  all  cases  fibrous  when  fractured. 


COLD    CRYSTALLIZATION   OF  IRON  AND    STEEL.       503 

In  the  report  is  said  :  — 

"  There  are  no  data  at  command  telling  what  the  original 
qualities  of  the  metal  of  these  bars  were  :  it  is  sufficient,  how- 
ever, to  find  toughness  and  a  fibrous  appearance  in  the  iron  to 
prove  that  brittleness  or  crystallization  has  not  resulted  from 
long  exposure  to  the  stresses  and  vibrations  these  bars  have 
sustained." 

The  only  other  evidence  that  will  be  referred  to  is  the  paper 
of  Mr.  A.  F.  Hill  upon  the  "  Crystallization  of  Iron  and  Steel," 
contained  in  the  Proceedings  of  the  Society  of  Arts  of  the 
Massachusetts  Institute  of  Technology  for  1882-83.  In  this 
article  Mr.  Hill  covers  the  ground  very  fully,  and  distinctly 
asserts  that  — 

"  The  fact  is  that  there  is  at  present  not  a  single  well- 
authenticated  instance  of  iron  or  steel  ever  having  become 
crystallized  from  use  under  temperatures  below  900°  F." 

He  claims  to  have  investigated  a  great  many  cases  where 
cold  crystallization  has  been  claimed,  and  to  have  found,  in 
every  case  where  crystals  existed,  that  at  some  period  of  its 
manufacture  or  working  the  metal  was  overheated.  He 
says : — 

"  That  the  crystalline  appearance  of  a  fracture  is  not  neces- 
sarily an  indication  of  the  presence  of  genuine  crystals  is  proven 
by  the  well-known  fact  that  a  skilful  blacksmith  can  fracture 
fine  fibrous  iron  or  steel  in  such  a  manner  as  to  let  it  appear 
either  fibrous  and  silky,  or  coarse  and  crystalline,  according  to 
his  method  of  breaking  the  bar.  On  the  other  hand,  where 
there  is  genuine  crystallization,  no  skill  of  manipulation  will 
avail  to  hide  that  fact  in  the  fracture.  The  most  striking 
illustrations  of  this  that  have  come  under  my  notice  are  the 
fractures  of  the  beam-strap  of  the  Kaaterskill,  and  of  the 
connecting-rod  of  the  chain-cable  testing-machine  at  the  Wash- 
ington Navy  Yard.  The  photographs  of  both  fractures  are 
submitted  to  you,  and  the  similarity  of  their  appearance  is 


5O4  APPLIED    MECHANICS. 

most  singular.  Yet  what  a  difference  in  the  development  of 
the  longitudinal  sections  by  acid  treatment,  which  are  also 
presented  to  you. 

"  In  the  Kaaterskill  accident  the  fractures  of  both  the 
upper  and  lower  arms  of  the  strap  were  found  to  be  short  and 
square.  The  appearance  of  the  fractured  faces  showed  no 
trace  of  fibre,  and  was  altogether  granular.  Yet  the  longitudi- 
nal section,  taken  immediately  through  the  break,  and  devel- 
oped by  acid  treatment,  shows  the  presence  of  but  few  and 
small  crystals,  and  the  generally  fibrous  character  of  the  iron 
used  in  the  strap. 

"  In  the  connecting-rod  of  the  chain-cable  testing-machine 
we  find  the  crystalline  appearance  of  the  fracture  less,  if  any- 
thing, than  that  of  the  beam-strap,  while  the  development  of 
the  longitudinal  section  by  acid  treatment  reveals  most  beauti- 
fully, in  this  case,  the  thoroughly  crystalline  character  of  the 
metal.  As  is  well  known,  this  rod,  after  many  years  of  service, 
finally  broke  under  a  comparatively  light  strain,  and  having  all 
along  been  supposed  to  have  been  carefully  made,  and  from 
well-selected  scrap,  its  intensely  crystalline  structure,  as  re- 
vealed by  the  fractures,  has  done  service  for  quite  a  number  of 
years  as  piece  de  resistance  in  all  the  '  cold-crystallization ' 
arguments  which  have  been  served  up  in  that  time." 

He  then  goes  on  to  say  that  he  cut  the  rod  in  a  longitudinal 
direction,  and  treated  the  section  with  acid  ;  that  some  of  the 
crystals  shown  are  so  large  as  to  be  discernible  with  the  naked 
eye  ;  that  the  treated  section  furnished  incontrovertible  evi- 
dence that 'the  rod,  aside  from  the  fact  of  being  badly  dimen- 
sioned anyhow,  was  made  of  poor  material,  badly  heated,  and 
msufrlciently  hammered,  all  records,  suppositions,  ^and  asser- 
tfons  to  the  contrary  notwithstanding ;  that  there  are  a  large 
number  of  crystals  composed  of  a  substance,  presumably  a 
ferro-carbide,  which  is  not  soluble  in  nitric  acid,  and  is  found 
in  steel  only  ;  that  the  deduction  from  the  large  amount  of  this 


COLD    CRYSTALLIZATION   OF  IRON  AND    STEEL.       505 

substance  is  that  the  pile  was  formed  of  rather  poorly  selected 
scrap,  with  steel  scrap  mixed  in  ;  that  evidences  of  bad  heating 
are  abundant  throughout ;  and  that  the  strongest  evidence 
against  the  presumption  that  these  crystals  were  formed  during 
the  service  of  the  rod,  or  while  the  metal  was  cold,  is  found  in 
the  groupings  of  the  crystals  during  their  formation,  as  shown 
in  the  tracing  developed  by  the  acid  ;  that  they  are  not  of  the 
same  chemical  composition,  the  lighter  parts  containing  much 
more  carbon  than  the  darker  ones ;  it  is  therefore  pretty  evi- 
dent that  with  the  grouping  of  the  crystals  a  segregation  of 
like  chemical  compounds  took  place,  and  this  of  course  would 
have  been  impossible  in  the  solid  state.  He  then  cites  an 
experiment  he  made,  in  which  he  took  a  slab  of  best  selected 
scrap  weighing  about  200  pounds  and  forged  it  down  to  a 
3-inch  by  3-inch  square  bar,  one-half  being  properly  forged, 
and  the  other  half  being  exposed  to  a  sharp  flame  bringing  it 
quickly  to  a  running  heat,  keeping  it  at  this  heat  some  time, 
and  then  hammering  lightly  and  then  treating  it  a  second  time 
in  a  similar  manner  ;  the  result  being,  that  while  no  difference 
was  discernible  in  the  appearance  of  the  two  portions,  when 
cut  and  treated  with  acid  the  portion  that  was  properly  made 
showed  itself  to  be  a  fair  representative  of  the  best  quality  of 
iron,  while  in  the  other  portion  the  crystallization  was  strongly 
marked,  the  majority  of  the  crystals  being  large  and  well 
developed. 

He  also  says  :  — 

"  The  fact  is,  all  hammered  iron  or  steel  is  more  or  less 
crystalline,  the  lesser  or  greater  degree  of  crystallization  de- 
pending altogether  upon  the  greater  or  lesser  skill  employed 
in  working  the  metal,  and  also  largely  upon  the  size  of  the 
forging.  Crystallization  tends  to  lower  very  sensibly  the  elastic 
limit  of  iron  and  steel,  and  therefore  hastens  the  deterioration 
of  the  metal  under  strain.  It  is  for  this  reason  that  large  a:id 
heavy  forgings  ought  to  be,  and  measurably  are,  excluded  as 


5O6  APPLIED    MECHANICS. 

much  as  possible  from  permanent  structures.  In  machine  con- 
struction we  cannot  do  without  them,  and  must  therefore 
accept  the  necessity  of  replacing  more  or  less  frequently  the 
parts  doing  the  heaviest  work." 

The  evidence  given  above  seems  to  the  writer  to  be  suffi- 
cient, and  to  warrant  the  conclusions  stated  on  pages  475,  476. 

EFFECT    OF    TEMPERATURE     UPON     THE     RESISTING    PROPER- 
TIES  OF   IRON   AND    STEEL. 

Much  the  best  and  most  systematic  work  upon  this  subject 
has  been  done  at  the  Watertown  Arsenal,  and  an  account  of  it 
is  to  be  found  in  "  Notes  on  the  Construction  of  Ordnance, 
No.  50,"  published  by  the  Ordnance  Department  at  Washing- 
ton, D.  C,  U.S.A. 

Other  references  are  the  following:  — 

Sir  William  Fairbairn:  Useful  Information  for  Engineers. 
Committee  of  Franklin  Institute:  Franklin  Institute  Journal. 
Knutt  Styffe  and  Christer  P.  Sandberg:  Iron  and  Steel. 
Kollman:  Engineering,  July  30,  1880. 
Massachusetts  R.  R.  Commissioners'  Report  of  1874. 
Bauschinger:  Mittheilungen,  Heft  13,  year  1886. 

A  summary  of  the  Watertown  tests,  largely  quoted  from 
the  above-mentioned  report,  will  be  given  here,  and  then  a  few 
remarks  will  suffice  for  the  others. 

The  subjects  upon  which  experiments  were  made  at  Water 
town  were  the  effect  of  temperatures  upon  — 

i°.  The  coefficient  of  expansion. 

2°.  The  modulus  of  elasticity. 

3°.  The  tensile  strength. 

4°.  The  elastic  limit. 

5°.  The  stress  per  square  inch  of  ruptured  section- 

6°.  The  percentage  contraction  of  area. 

7°.  The  rate  of  flow  under  stress. 

8°.  The  specific  gravity. 


EPFECT  OF   TEMPERATURE   ON  TRON  AND    STEEL.      S°7 


9°.  The  strength  when  strained  hot  and  subsequently  rup- 
tured cold. 

10°.  The  color  after  cooling. 
11°.  Riveted  joints. 

1°.    THE    COEFFICIENTS    OF    EXPANSION. 

These  were  determined  from  direct  measurements  upon  the 
experimental  bars,  first  measuring  their  lengths  on  sections 
35  inches  long,  while  the  bars  were  immersed  in  a  cold  bath  of 
ice-water,  and  again  measuring  the  same  sections  after  a  period 
of  immersion  in  a  bath  of  hot  oil. 

The  range  of  temperature  employed  was  about  210  degrees 
Fahr.,  as  shown  by  mercurial  thermometers. 

Observations  were  repeated,  and  again  after  the  steel  bars 
had  been  heated  and  quenched  in  water  and  in  oil. 

The  average  values  are  exhibited  in  the  following :  — 

TABLE   I. 
First  Series  of  Bars. 


Metal. 

Chemical  Composition. 

Coefficients    of     Expansion 
per    Degree    Fahr.,    per 
Unit  of  Length. 

C. 

Mn. 

Si. 

Wrought-iron. 

.0000067302 

Steel. 

.09 

.  II 

.0000067561 

" 

.20 

•45 

.0000066259 

" 

•31 

•57 

.0000065149 

•37 

.70 

.0000066597 

4  ' 

.51 

•58 

.02 

.OOOOO662O2 

tt 

•57 

•93 

.07 

.0000063891 

" 

•  71 

•58 

.08 

.0000064716 

" 

.81 

•56 

•17 

.0000062167 

11 

.89 

•57 

.19 

.0000062335 

*  * 

•97 

.80 

.28 

.0000061700 

Cast-  (gun)  iron. 

.0000059261 

Drawn  copper. 

0000091286 

APPLIED    MECHANICS 


Subsequent  determinations  of  the  coefficient  of  expansion 
of  a  second  series  of  steel  bars  gave  — 


TABLE  II. 


Jnemical  L 

omposition 

Coefficients  of  Expansion 

c. 

Mn. 

Si. 

S. 

P. 

Cu. 

per  Degree  Fahr.,  per 
Unit  of  Length. 

•  17 

I.I3 

.023 

.122 

.079 

.04 

.0000067886 

.20 

.69 

•037 

•13 

.078 

.26 

.0000068567 

.21 

.26 

.08 

.14 

-.059 

.00 

.0000067623 

.26 

.07 

.11 

.096 

.08 

•  047s 

.0000067476 

.26 

.26 

.07 

.112 

.06 

.038 

.0000067102 

.26 

.28 

.07 

."5 

.062 

•035 

.0000067175 

.28 

•23 

.09 

.168 

.09 

.178 

.0000067794 

•43 

•97 

•05 

.08 

.096 

.024 

.0000066124 

•43 

i.  08 

•  037 

.08 

.114 

•233 

.0000066377 

•53 

•75 

.10 

.078 

.087 

.174 

.0000064181 

•55 

1.02 

.05 

.078 

.12 

•  15 

.OOOOo66l22 

.72 

.70 

.18 

.07 

•13 

•23 

.0000064330 

.72 

.76 

.20 

.056 

.086 

.186 

.  0000063080 

•79 

.86 

.21 

.084 

•093 

.096 

.0000063562 

.07 

.07 

•13 

.01 

.018 

.006 

.0000061528 

.08 

.12 

.I9 

.Oil 

.02 

trace 

.0000061702 

.12 

.10 

.09 

.013 

.018 

trace 

.0000060716 

.14 

.IO 

•15 

trace 

.018 

trace 

.0000062589 

•17 

.10 

.10 

trace 

.018 

o 

.0000061332 

•31 

•13 

.19 

.Oil 

.026 

trace 

.0000061478 

Ten  bars  of  the  first  series  were  now  heated  a  bright  cherry- 
red  and  quenched  in  oil  at  80°  Fahr.,  the  hot  bars  successively 
raising  the  temperature  of  the  oil  to  about  240°  Fahr.,  the  bath 
being  cooled  between  each  immersion. 

The  behavior  of  the  bars  under  rising  temperature,  when 
examined  for  coefficients  of  expansion,  seemed  somewhat 
erratic,  the  highest  temperature  reached  being  235°  ;  but  this 
behavior  was  subsequently  explained  by  the  permanent  changes 
in  length  found  when  the  bars  were  returned  to  the  cold  bath. 


EFFECT  OF    TEMPERATURE   ON  IRON  AND   STEEL.     $09 

Generally  the  bars  were  found  permanently  shortened  at  the 
close  of  these  observations. 

The  bars  were  again  heated  bright  cherry-red  and  quenched 
in  water  at  50°  to  55°  Fahr.,  the  water  being  raised  by  the 
quenching  to  1 10°  to  125°  Fahr. 

After  resting  72  hours,  measurements  were  taken  in  the  cold 
bath,  followed  by  a  rest  of  18  hours,  when  they  were  heated 
and  measured  in  the  hot  bath,  after  which  they  were  measured 
in  the  cold  bath ;  the  maximum  temperature  reached  with  the 
hot  bath  being  233°. 7  Fahr.,  erratic  behavior  occurring  still. 

They  were  next  heated  in  an  oil  bath  at  300°  Fahr.,  and 
kept  at  this  temperature  6  hours,  then  cooled  in  the  bath  ;  15 
hours  later  they  were  heated  to  243°  Fahr.,  and  again  measured 
hot,  and  then  cold.  These  downward  readings  showed  the 
quenched  in  water  bars  to  have  their  coefficients  elevated 
above  the  normal,  as  shown  in  the  following  table,  these 
being  the  same  steel  bars  as  in  Table  I,  and  in  the  sarr.e 
order  :  — 

TABLE    III. 


Coefficients  of    Expansion 

Apparent   Shortening  of  Bars 
Due  to  Six    Hours  at   300° 

per  Degree  per  Unit  of 
Length. 

Fahr.,    and     the    following 
Immersion  in  the  Hot  Bath. 

.0000067641 

—  .0006 

.0000066622 

.0002 

.0000066985 

.OOl6 

.0000067377 

.OO23 

.0000069776 

—   .0004 

.0000067041 

.0082 

.0000066939 

.0064 

.0000068790 

.0054 

.0000072906 

.0055 

.0000071578 

.0048 

510  APPLIED   MECHANICS. 

Finally  the  bars  were  annealed  by  heating  bright  red  and 
cooling  in  pine  shavings,  the  effect  of  which  was  to  approxi- 
mately restore  the  rate  of  expansion  to  the  normal,  as  shown 
by  Table  I  for  these  ranges  of  temperature. 

2°.    MODULUS    OF    ELASTICITY. 

These  were  obtained  with  the  first  series  of  bars  at  atmos^ 
pheric  temperatures,  and  at  higher  temperatures,  up  to  495° 
Fahr. 

There  occurred  invariably  a  decrease  in  the  modulus  of 
elasticity  with  an  increase  in  temperature,  and,  in  the  case  of 
the  specimens  tested,  the  low  carbon  steels  showed  a  greater 
reduction  in  the  modulus  than  the  high  carbon  steels,  the 
first  specimen  having  a  modulus  of  elasticity  at  the  minimum 
temperature  30612000,  and  at  the  maximum  27419000,  while 
the  last  specimen  had  at  the  minimum  temperature  29126000, 
and  at  the  maximum  27778000. 

3°.    TENSILE   STRENGTH. 

The  tests  were  made  upon  the  first  series  of  steel  bars, 
wrought-irons  marked  A  and  B,  a  muck-bar  railway  axle,  and 
cast-iron  specimens  from  a  slab  of  gun-iron. 

The  specimens  were  o".798  diameter,  and  5"  length  of  stem, 
having  threaded  ends  \fl '.25  diameter. 

Wrought-iron  A  was  selected  because  it  was  found  very  hot 
short  at  a  welding  temperature.  It  had  been  strained  with  a 
tensile  stress  of  42320  pounds  per  square  inch  seven  years 
previous  to  being  cut  up  into  specimens  for  the  hot  tests. 

The  specimens  while  under  test  were  confined  within  a 
sheet-iron  muffle,  through  the  ends  of  which  passed  auxiliary 
bars  screwed  to  the  specimens,  the  auxiliary  bars  being  secured 
to  the  testing-machine. 


EFFEC7"   OF   TEMPERATURE   ON  IRON  AND    STEEL.    $11 

The  heating  was  done  by  means  of  gas-burners  arranged 
below  the  specimen  and  within  the  muffle. 

The  temperature  of  the  test-bar  was  estimated  from  the 
expansion  of  the  metal,  observed  on  a  specimen  length  of  six 
inches,  using  the  coefficients  which  were  determined  at  lower 
temperatures,  as  hereinbefore  stated,  assuming  there  was  a 
uniform  rate  of  expansion. 

Access  to  the  specimen  for  the  purpose  of  measuring  the 
expansion  was  had  through  holes  in  the  top  of  the  muffle. 
The  temperature  was  regulated  by  varying  the  number  of  gas- 
burners  in  use,  the  pressure  of  the  gas,  and  also  by  means  of 
diaphragms  placed  within  the  muffle  for  diffusing  the  heat. 

The  approximate  elongations  under  different  stresses  were 
determined  during  the  continuance  of  a  test  from  measurements 
made  on  the  hydraulic  holders  of  the  testing-machine,  at  a 
convenient  distance  from  the  hot  muffle,  correcting  these 
measurements  from  data  obtained  by  simultaneous  micrometer 
readings  made  on  the  specimen  and  the  hydraulic  holders  at 
atmospheric  temperatures. 

While  it  does  not  seem  expedient  in  one  series  of  tests  to 
obtain  complete  results  upon  the  tensile  properties  at  high 
temperatures,  yet,  incidentally,  much  additional  valuable  infor- 
mation may  be  obtained  while  giving  prominence  to  one  or 
more  features. 

From  these  elongations  the  elastic  limits  were  established 
where  the  elongations  increased  rapidly  under  equal  incre- 
ments of  load.  Proceeding  with  the  test  until  the  maximum 
stress  was  reached,  recorded  as  the  tensile  strength,  observing 
the  elongation  at  the  time,  then,  when  practicable,  noting 
the  stress  at  the  time  of  rupture." 

For  the  detailed  tables  of  tests  the  student  is  referred  to 
the  "  Notes  on  the  Construction  of  Ordnance." 

The  elastic  limits  and  tensile  strengths  are  computed  in 
pounds  per  square  inch,  both  on  original  sectional  areas  of  the 


APPLIED    MECHANICS. 


specimens  and  on  the  minimum  or  reduced  sections,  as  meas- 
ured at  the  close  of  the  hot  tests. 

From  the  results  it  appears  that  the  tensile  strength  of  the 
steel  bars  diminishes  as  the  temperature  increases  from  zero 
Fahr.,  until  a  minimum  is  reached  between  200°  and  300°  Fahr., 
the  milder  steels  appearing  to  reach  the  place  of  minimum 
strength  at  lower  temperatures  than  the  higher  carbon  bars. 

From  the  temperature  of  this  first  minimum  strength  the 
bars  display  greater  tenacity  with  increase  of  temperature,  until 
the  maximum  is  reached  between  the  temperatures  of  about 
400°  to  650°  Fahr. 

The  higher  carbon  steels  reach  the  temperature  of  maximum 
strength  abruptly,  and  retain  the  highest  strength  over  a  lim- 
ited range  of  temperature.  The  mild  steels  retain  the  increased 
tenacity  over  a  wider  range  of  temperature. 

From  the  temperature  of  maximum  strength  the  tenacity 
diminishes  rapidly  with  the  high  carbon  bars,  somewhat  less 
so  with  mild  steels,  until  the  highest  temperatures  are  reached, 
covered  by  these  experiments. 

The  greatest  loss  observed  in  passing  from  70°  Fahr.  to  the 
temperature  of  first  minimum  strength  was  6.5  per  cent  at 
295°  Fahr. 

The  greatest  gain  over  the  strength  of  the  metal  at  70°  was 
25.8  per  cent  at  460°  Fahr. 

The  several  grades  of  metal  approached  each  other  in 
tenacity  as  the  higher  temperatures  were  reached.  Thus  steels 
differing  in  tensile  strength  nearly  90000  pounds  per  square 
inch  at  70°,  when  heated  to  1600°  Fahr.  appear  to  differ  only 
about  loooo  pounds  per  square  inch. 

The  rate  of  speed  of  testing  which  may  modify  somewhat 
the  results  with  ductile  material  at  atmospheric  temperatures 
has  a  very  decided  influence  on  the  apparent  tenacity  at  high 
temperatures. 

A  grade   of   metal  which,  at  low  temperatures,  had  little 


EFFECT  OF  TEMPERATURE   ON  IRON  AND   STEEL.     $13 

ductility,  displayed  the  same  strength  whether  rapidly  or  slowly 
fractured  from  the  temperature  of  the  testing-room  up  to  600° 
Fahr. ;  above  this  temperature  the  apparent  strength  of  the 
rapidly  fractured  specimens  largely  exceeded  the  others. 

At  1410°  Fahr.  the  slowly  fractured  bar  showed  33240 
pounds  per  square  inch  tensile  strength,  while  a  bar  tested  in 
two  seconds  showed  63000  pounds  per  square  inch. 

Cast-iron  appeared  to  maintain  its  strength  with  a  tendency 
to  increase  until  900°  Fahr.  is  reached,  beyond  which  the 
strength  diminishes.  Under  the  higher  temperatures  it  devel- 
oped numerous  cracks  on  the  surface  of  the  specimens  preced- 
ing complete  rupture. 

4°.    ELASTIC    LIMIT. 

The  report  says  of  this  that  it  appears  to  diminish  with  in- 
crease of  temperature.  Owing  to  a  period  of  rapid  yielding  with- 
out increase  of  stress,  or  even  under  reduced  stress,  the  elastic 
limit  is  well  defined  at  moderate  temperatures  with  most  of  the 
steels. 

Mild  steel  shows  this  yielding  point  up  to  the  vicinity  of 
500°;  in  hard  steels,  if  present,  it  appears  at  lower  temperatures. 

The  gradual  change  in  the  rate  of  elongation  at  other  times 
often  leaves  the  definition  of  the  elastic  limit  vague  and  doubt- 
ful, especially  so  at  high  temperatures.  The  exclusion  of  de- 
terminable  sets  would  in  most  cases  place  the  elastic  limit  below 
the  values  herein  given. 

In  approaching  temperatures  at  which  the  tensile  proper- 
ties are  almost  eliminated  exact  determinations  are  correspond- 
ingly difficult,  the  tendency  being  to  appear  to  reach  too  high 
values. 

5°.    STRESS    ON    THE    RUPTURED    SECTION. 

This,  generally,  follows  with  and  resembles  the  curve  of 
tensile  strength. 


514  APPLIED    MECHANICS. 

Specimens  of  large  contraction  of  area,  tested  at  high 
temperature,  have  given  evidence  on  the  fractured  ends  of 
having  separated  at  the  centre  of  the  bar  before  the  outside 
metal  parted. 

Elongation  under  Stress^ 

Although  the  metal  is  capable  of  being  worked  under  the 
hammer  at  high  temperatures,  it  does  not  then  possess  sufficient 
strength  within  itself  to  develop  much  elongation,  general 
elongation  being  greatest  at  lower  temperatures. 

Greater  rigidity  exists  under  certain  stresses  at  intermedi- 
ate temperatures  than  at  either  higher  or  lower  temperatures. 

Thus  one  of  the  specimens  tested  at  569°  Fahr.  showed  less 
elongation  under  stresses  above  50000  pounds  per  square  inch 
than  the  bars  strained  at  higher  or  lower  temperatures. 

Two  other  specimens  showed  a  similar  behavior  at  315°  and 
387°  respectively,  and  likewise  other  specimens. 

In  bars  tested  at  about  200°  to  400°  Fahr.  there  are  dis- 
played alternate  periods  of  rigidity  and  relaxation  under  in- 
creasing stresses,  resembling  the  yielding  described  as  occur- 
ring with  some  bars  immediately  after  passing  the  elastic 
limit. 

The  repetition  of  these  intervals  of  rigidity  and  relaxation 
is  suggestive  of  some  remarkable  change  taking  place  within 
the  metal  in  this  zone  of  temperature. 

6°.    PERCENTAGE    CONTRACTION    OF   AREA. 

This  varies  with  the  temperature  of  the  bar  ;  it  is  somewhat 
less  in  mild  and  medium  hard  steels  at  400°  to  600°  than  at 
atmospheric  temperatures. 

Above  500°  or  600°  the  contraction  increases  with  the 
temperature  of  the  metal ;  with  three  exceptions,  which  showed 
diminished  contraction  at  1100°  Fahr.,  until  at  the  highest 
temperatures  some  of  them  were  drawn  down  almost  to  points. 


EFFECT  OF   TEMPERATURE   ON  IRON  AND    STEEL. 


7°.    RATE    OF    FLOW    UNDER    STRESS. 

The  full  effect  of  a  load  superior  to  the  elastic  limit  is  not 
immediately  felt  in  the  elongation  of  a  ductile  metal,  and  the 
same  is  true  at  higher  temperatures. 

The  flow  caused  by  a  stress  not  largely  in  excess  of  the 
elastic  limit  has  a  retarding  rate  of  speed,  and  eventually  ceases 
altogether  ;  whereas  under  a  high  stress  the  rate  of  flow  may 
accelerate,  and  end  in  rupture  of  the  metal. 

Hence  the  apparent  tensile  strength  maybe  modified  within 
limits  by  the  time  employed  in  producing  fracture. 

8°.    SPECIFIC    GRAVITY. 

In  general,  the  specific  gravity  is  materially  diminished  in 
the  vicinity  of  the  fractured  ends  of  tensile  specimens,  and  this 
diminution  takes  place  in  the  different  grades  of  steel,  in  bars 
ruptured  under  different  conditions  of  temperature,  stress,  and 
contraction  of  area. 


9       BARS    STRAINED    HOT,    AND    SUBSEQUENTLY    RUPTURED    COLD. 

The  effect  of  straining  hot  on  the  subsequent  strength  cold 
appears  to  depend  upon  the  magnitude  of  the  straining  force 
and  the  temperature  in  the  first  instance. 

There  is  a  zone  of  temperature  in  which  the  effect  of  hot 
straining  elevates  the  elastic  limit  above  the  applied  stress,  and 
above  the  primitive  value,  and  if  the  straining  force  approaches 
the  tensile  strength,  there  is  also  a  material  elevation  of  that 
value  when  ruptured  cold.  These  effects  have  been  observed 
within  the  limits  of  about  335°  and  740°  Fahr. 

After  exposure  to  higher  temperatures  there  occurs  a 
gradual  loss  in  both  the  elastic  limit  and  tensile  strength,  and 
generally  a  noticeable  increase  in  the  contraction  of  area. 


5l6  APPLIED    MECHANICS. 


This  was  not  sensibly  changed  by  temperatures  below  200°. 
After  300°  the  metal  was  light  straw-colored  :  after  400°,  deep 
straw ;  from  500°  to  600°,  purple,  bronze-colored,  and  blue  ; 
after  700°,  dark  blue  and  blue  black. 

After  800°  some  specimens  still  remained  dark  blue.  After 
heating  above  about  800°  the  final  color  affords  less  satisfactory 
means  of  approximately  judging  of  the  temperature,  the  color 
remaining  a  blue  black,  and  darker  when  a  thick  magnetic 
oxide  is  formed. 

At  about  1100°  the  surface  oxide  reaches  a  tangible  thick- 
ness, a  heavy  scale  of  o".ooi  to  o".oo2  thickness  forming  as 
higher  temperatures  are  reached.  The  red  oxide  appears  at 
about  1500°. 

11°.    IN    THE    TESTS   OF    RIVETED   JOINTS 

of  steel  boiler-plates  at  temperatures  ranging  from  70°  to  about 
700°  Fahr.  the  indications  of  the  tensile  tests  of  plain  bars  were 
corroborated. 

Joints  at  200°  Fahr.  showed  less  strength  than  when  cold ; 
at  250°  and  higher  temperatures  the  strength  exceeded  the 
cold  joints ;  and  when  overstrained  at  400°  and  500°  there  was 
found,  upon  completing  the  test  cold,  an  increase  in  strength. 

Rivets  which  sheared  cold  at  40000  to  41000  pounds  per 
square  inch,  at  300°  Fahr.  sheared  at  46000  pounds  per  square 
inch ;  and  at  600°  Fahr.,  the  highest  temperature  at  which  the 
joints  failed  in  this  manner,  the  shearing-strength  was  42130 
pounds  per  square  inch. 

In  addition  to  the  work  at  Watertown  which  has  just  been 
detailed  two  other  matters  will  be  referred  to  here. 


EFFECT  OF  TEMPERATURE   ON  IRON  AND   STEEL.     $1? 

1°.  It  is  well  known  that  wrought-iron  and  steel  are  very 
brittle  at  a  straw  heat  and  a  pale  blue,  as  shown  by  the  fact  that 
when  the  attempt  is  made  to  bend  a  specimen  at  these  tempera- 
tures it  results  in  cracking  it  some  time  before  a  complete  bend- 
ing can  be  effected,  even  in  the  case  of  metal  which  is  so  ductile 
that  it  can  be  bent  double  cold,  red  hot,  or  at  a  flanging  heat, 
without  showing  any  signs  of  cracking. 

2°.  Bauschinger  defines  the  elastic  limit  as  the  load  at  which 
the  stress  is  no  longer  proportional  to  the  strain  ;  whereas  he 
calls  stretch-limit  (Streckgrenze)  the  load  at  which  the  strain 
diagram  makes  a  sudden  change  in  its  direction  ;  i.e.,  where 
instead  of  showing  a  gradually  increasing  ratio  of  strain  to  stress 
it  shows  a  sudden  and  rapid  increase. 

From  his  experiments  (see  Heft  13  of  the  Mittheilungen, 
year  1886)  he  draws  the  following  conclusions  :  — 

(a)  That  the  effect  of  heating  and  subsequent  cooling  in 
lowering  both  the  elastic  and  the  stretch  limits  in  mild   steel 
begins  at  about  660°  Fahr.  when  the  cooling  is  sudden,  and  at 
about  840°  Fahr.  when  it  is  slow,  and  for  wrought-iron  at  about 
750°  with  either  rapid  or  slow  cooling. 

(b)  That  the  operation  of  heating  above  those  temperatures, 
and  of  subsequent  slow  or  quick  cooling,  is  that  both  the  elastic 
and  the  stretch  limit  are  lowered,  and  the  more  so  the  greater 
the  heating ;  also,  that  this  effect  is  greater  on  the  elastic  than 
on  the  stretch  limit. 

(c)  Quick  cooling  after  heating  higher  than  the  above-stated 
temperatures  lowers  the  elastic  and  the  stretch  limit,  especially 
the  first,  much  more  than  slow  cooling,  dropping  the  elastic 
limit  almost  immediately  at  a  heat  of  about  930°  and  certainly 
at  a  red  heat  to  nothing  or  nearly  nothing  in  wrought-iron,  and 
in  both  mild  and  hard  steel,  while  slow  cooling  cannot  bring 
about  such  a  great  drop  of  the  elastic  limit,  even  from  more 
than  a  red  heat. 


APPLIED   MECHANICS. 


Effect  of  Cold-Rolling  on  Iron  and  Steel. — It  has  already 
been  stated,  p.  410,  that  it  was  discovered  independently  by 
Commander  Beardslee  and  Professor  Thurston,  that  if  a  load 
were  gradually  applied  to  a  piece  of  iron  or  steel  which  exceeded 
its  elastic  limit,  and  the  piece  then  allowed  to  rest,  the  elastic 
limit  and  the  ultimate  strength  would  thus  be  increased.  This 
may  be  accomplished  with  soft  iron  and  steel  by  cold-rolling  or 
cold-drawing,  but  cannot  be  taken  advantage  of  in  hard  iron 
or  steel. 

Professor  Thurston,  who  has  investigated  this  matter  at 
great  length,  and  made  a  large  number  of  tests  on  the  subject, 
gives  the  following  as  the  results  of  cold-rolling:  — 


Increase  in 

Per  Cent. 

Tenacity  

2C  to     4.0 

Transverse  stress  ...«. 

CQ  to    80 

Elastic  limit  (tension,  torsion,  and  transverse), 

80  to  125 
300  to  400 

Elastic  resilience  (transverse) 

I  CO  to  A.2  C 

He  also  says,  in  regard  to  the  modulus  of  elasticity,  — 

"  Collating  the  results  of  several  hundred  tests,  the  author 
[Professor  Thurston]  found  that  the  modulus  of  elasticity  rose, 
in  cold-rolling,  from  about  25000000  Ibs.  per  square  inch  to 
26000000,  the  tenacity  .from  52000  Ibs.  to  nearly  70000,  the 
elastic  limit  from  30000  Ibs.  to  nearly  60000  Ibs.  ;  and  the  ex- 
tension was  reduced  from  25  to  ioj  per  cent. 

"  Transverse  loads  gave  a  reduction  of  the  modulus  of  elas- 
ticity to  the  extent  of  about  1000000  Ibs.  per  square  inch,  an 
increase  in  the  modulus  of  rupture  from  73600  to  133600,  and 
reduction  of  deflection  at  maximum  load  of  about  25  per  cent. 
The  resistance  of  the  elastic  limit  was  doubled,  and  occurred 
at  a  much  greater  deflection  than  with  untreated  iron." 

On  the  other  hand,  the  two  steel  eye-bars  referred  to  on 


FACTOR    OF  SAFETY. 


519 


p.  472  show  a  decrease  of  modulus  of  elasticity  with  increasing 
overstrain. 

Whitworth's  Compressed  Steel. — Sir  Joseph  Whitworth  pro- 
duces steel  of  great  strength  by  applying  to  the  molten  metal, 
directly  after  it  leaves  the  furnace,  a  pressure  of  about  14000 
Ibs.  per  square  inch;  this  being  sufficient  to  reduce  the  length 
of  an  eight-foot  column  by  one  foot.  He  claims,  according  to 
D.  K.  Clark,  to  be  able  to  obtain  with  certainty  a  strength  of 
40  English  tons  with  30  per  cent  ductility,  and  mild  steel  of  a 
strength  of  30  English  tons  with  33  or  34  per  cent  ductility. 

The  following  tests  were  made  on  the  Watertown  machine, 
upon  some  specimens  of  Whitworth  steel  taken  from  a  section 
of  a  jacket  which  was  shrunk  upon  a  wrought-iron  tube,  and 
removed  from  shrinkage  by  the  application  of  high  furnace  heat : 

TENSILE  TESTS. 


Diameter, 
Inches. 

Tensile 
Strength, 
Ibs.  per  Sq.  In. 

Elastic  Limit, 
Ibs.  per  Sq.  In. 

Contraction 
of  Area, 
per  cent, 

o  564 

103960 

55000 

41.9 

0.564 

90040 

48000 

47.2 

0.564 

104200 

57000 

24.6 

0.564 

IOOI20 

57000 

44.6 

0.564 

93040 

53000 

39-2 

0.564 

104160 

60000 

24.6 

0.564 

93160 

47000 

39-2 

COMPRESSIVE   TESTS. 


Length, 
Inches. 

Diameter, 
Inches. 

Compressive 
Strength, 
Ibs.  per  Sq.  In. 

Elastic  Limit, 
Ibs.  per  Sq.  In. 

5 

0.798 

IO2IOO 

61000 

5 

0.798 

89000 

57000 

3-94 

0.798 

IOI6OO 

53000 

3-94 

0.798 

IOI6OO 

54000 

§  227.    Factor   of    Safety In    order   to   determine   the 

proper  dimensions  of  any  loaded  piece,  it  becomes  necessary 


52O  APPLIED   MECHANICS. 

to  fix,  in  some  way,  upon  the   greatest   allowable   stress   per 
square  inch  to  which  the  piece  shall  be  subjected. 

The  most  common  practice  has  been  to  make  this  some 
fraction  of  the  breaking-strength  of  the  material  per  square 
inch. 

As  to  how  great  this  factor  should  be,  depends  upon  — 

i°.  The  use  to  which  the  piece  is  to  be  subjected ; 

2°.  The  liability  to  variation  in  the  quality  of  the  material ; 

3°.  The  question  whether  we  are  considering,  as  the  load 
upon  the  piece,  the  average  load,  or  the  greatest  load  that  can 
by  any  possibility  come  upon  it ; 

4°.  The  question  as  to  whether  the  structure  is  a  temporary 
or  a  permanent  one; 

5°.  The  amount  of  injury  that  would  be  done  by  breakage 
of  the  piece ; 
and  other  considerations. 

The  factors  most  commonly  recommended  are,  3  for  a  dead 
or  quiescent  load,  and  6  for  a  live  or  moving  load. 

A  common  American  and  English  practice  for  iron  bridges 
is  to  use  a  factor  of  safety  of  4  for  both  dead  and  moving  load. 
In  machinery  a  factor  as  large  as  6  is  desirable  when  there  is 
no  liability  to  shocks  ;  and  when  there  is,  a  larger  factor  should 
be  used. 

A  method  sometimes  followed  for  tension  and  compression 
pieces  is,  to  prescribe  that  the  stretch  under  the  given  load 
should  not  exceed  a  certain  fixed  fraction  of  the  length.  This 
requires  a  knowledge  of  the  modulus  of  elasticity  of  the  mate- 
rial. 

In  the  case  of  a  piece  subjected  to  a  transverse  load,  it  is 
the  most  common  custom  to  determine  its  dimensions  in  accord- 
ance with  the  principle  of  providing  sufficient  strength  ;  and 
for  this  purpose  a  certain  fraction  (as  one-fourth)  of  the  mod- 
ulus of  rupture  is  prescribed  as  the  greatest  allowable  safe 
stress  per  square  inch  at  the  outside  fibre.  Thus,  for  wrought- 
iron  from  10000  to  12000  Ibs.  per  square  inch  is  often  adopted 


REPEATED    STRESSES.  $21 

as  the  greatest  allowable  stress  at  the  outside  fibre,  this  being 
about  one-fourth  of  the  modulus  of  rupture. 

The  other  method  for  dimensioning  a  beam  is,  to  prescribe 
its  stiffness ;  i.e.,  that  it  shall  not  deflect  under  its  load  more 
than  a  certain  fraction  of  the  span.  This  fraction  is  taken  as 

rb- to  7TT<7- 

This  latter  method  depends  upon  the  modulus  of  elasticity 

of  the  beam ;  and  while  it  is  the  most  advisable  method  to 
follow,  and  as  a  rule  would  be  safer  than  the  other  method, 
nevertheless,  in  the  case  of  very  stiff  and  brittle  material  it 
might  be  dangerous ;  hence  we  ought  to  know  also  the  break- 
ing-weight and  the  limit  of  elasticity  of  the  beam  we  are  to  use, 
and  not  allow  it  to  approach  either  of  these.  This  precaution 
will  be  especially  important  to  observe  in  the  case  of  steel 
beams,  which  are  only  now  being  introduced. 

On  the  other  hand,  in  moving  machinery  a  factor  of  safety 
of  six  is  usually  required  when  there  is  no  unusual  exposure  to 
shocks,  as  in  smooth-running  shafting,  etc. ;  and  when  there 
are  irregular  shocks  liable  to  come  upon  the  piece,  a  greater 
factor  is  used. 

WOHLER'S  RESULTS. 

§  228.  Repeated  Stresses. — The  extensive  experiments  of 
Wohler  for  the  Prussian  government,  which  were  subsequently 
carried  on  by  his  successor,  Spangenberg,  were  made  to  deter- 
mine the  effect  of  oft-repeated  stresses,  and  of  changes  of 
stress,  upon  wrought-iron  and  steel. 

In  the  ordinary  American  and  English  practice,  it  is  cus- 
tomary, in  determining  the  dimensions  of  a  piece,  as  of  a  bridge 
member,  to  ascertain  the  greatest  load  which  the  piece  can 
ever  be  called  upon  to  bear,  and  to  fix  the  size  of  the  piece  in 
accordance  with  this  greatest  load. 

Wohler  called  attention  to  the  fact  that  the  load  that  would 
break  a  piece  depends  upon  both  the  greatest  and  least  load 
that  it  would  ever  be  called  upon  to  bear.  Thus,  a  tension-rod 


522  APPLIED    MECHANICS. 

which  is  subjected  to  alternate  changes  of  load  extending  from 
20000  to  80000  Ibs.  would  require  a  greater  area  for  safety  than 
one  which  was  subjected  to  loads  varying  only  between  the 
limits  of  60000  and  80000  Ibs.  ;  and  this  would  require  more 
area  than  one  which  was  subjected  to  a  steady  load  of  80000 
Ibs. 

Wohler  expresses  this  law  as  follows,  in  his  "  Festigkeits 
versuche  mit  Eisen  und  Stahl." 

"The  law  discovered  by  me,  whose  universal  application 
for  iron  and  steel  has  been  proved  by  these  experiments,  is  as 
follows :  The  fracture  of  the  material  can  be  effected  by 
variations  of  stress  repeated  a  great  number  of  times,  of 
which  none  reaches  the  breaking-limit.  The  differences  of 
the  stresses  which  limit  the  variations  of  stress  determine  the 
breaking-strength.  The  absolute  magnitude  of  the  limiting 
stresses  is  only  so  far  of  influence  as,  with  an  increasing  stress, 
the  differences  which  bring  about  fracture  grow  less. 

"  For  cases  where  the  fibre  passes  from  tension  to  compres- 
sion and  vice  versa,  we  consider  tensile  strength  as  positive 
and  compressive  strength  as  negative ;  so  that  in  this  case  the 
difference  of  the  extreme  fibre  stresses  is  equal  to  the  greatest 
tension  plus  the  greatest  compression." 

Besides  the  ordinary  tests  of  tensile,  compressive,  shearing, 
and  torsional  strength,  he  made  his  experiments  mainly  on  the 
following  two  cases  :  — 

i°.  Repeated  tensile  strength;  the  load  being  applied  and 
wholly  removed  successively,  and  the  number  of  repetitions 
required  for  fracture  counted. 

2°.  Alternate  tension  and  compression  of  equal  amounts 
successively  applied,  the  number  of  repetitions  required  for 
fracture  being  counted. 

In  making  these  two  sets  of  tests,  he  made  the  first  set  in 
two  ways :  — 

(a)  By  applying  direct  tension. 


LAUNHARDT'S  FORMULA.  523 

(b)  By  applying  a  transverse  load,  and  determining  the 
greatest  fibre  stress. 

The  second  set  of  tests  was  made  by  loading  at  one  end  a 
piece  of  shaft  fixed  in  direction  at  the  other,  and  then  causing 
it  to  revolve  rapidly,  each  fibre  passing  alternately  from  tension 
to  an  equal  compression,  and  vice  versa. 

He  also  tried  a  few  experiments  where  the  lower  limit  of 
stress  was  neither  zero  nor  equal  to  the  upper  limit,  with  a 
minus  sign,  also  some  experiments  on  torsion,  on  shearing, 
and  on  repeated  torsion. 

When  Wohler  had  made  his  experiments,  and  published  his 
results,  there  were  a  number  of  attempts  made  by  different 
persons  to  deduce  formulae  which  should  depend  upon  these 
experiments  for  their  constants,  and  which  should  serve  to  deter- 
mine the  breaking-strength  for  any  given  variation  of  stresses. 

Only  two  of  these  formulae  will  be  given  here,  viz. : 

i°    That  of  Launhardt  for  one  kind  of  stress, 

2°   That  of  Weyrauch  for  alternate  tension  and  compression. 

LAUNHARDT'S  FORMULA. 

The  constants  used  in  this  formula  are : 

i°.  /,  the  carrying-strength  (Tragfestigkeit)  of  the  material 
per  unit  of  area,  which  is  the  same  as  the  tensile  strength  as 
determined  by  the  ordinary  tensile  testing-machine. 

2°.  u,  the  primitive  breaking-strength  (Ursprungsfestigkeit), 
i.e.,  the  greatest  stress  per  unit  of  area  of  which  the  piece  can  bear, 
without  breaking,  an  unlimited  number  of  repetitions,  the  load 
being  entirely  removed  between  times.  These  two  quantities 
have  been  determined  experimentally  by  Wohler;  and  it  is  the 
object  of  Launhardt's  formula  to  deduce,  in  terms  of  /,  u,  and  the 
ratio  between  the  greatest  and  least  loads  to  which  the  piece  is 
ever  subjected,  the  value  a  of  the  breaking-strength  per  unit  of 
area  when  these  loads  are  applied. 


524  APPLIED   MECHANICS. 


Let  the  greatest  stress  per  unit  area  be  a. 
the  least  stress  per  unit  area  be  c. 

Plot  the  values  of  -  as  abscissae,  and  those  of  a  as  ordinates, 
making   OA  =  u  (since  when  -  =  o,  a  =  u),  OC=i,  and    CB  =  t 

(since  when  -=i,  a  =  /).     Then  will  any  curve 

,          E     B       which  passes   through  the  points  A  and  B  have 
\\^~\~\        ^or  *ts  or(^mates  values  of  a  that  will  satisfy  the 


conditions  that  when  c  =  o,  a  —  u,  and  when  c  =  /, 
a  =  t.     By  assuming  for  this  curve,  the  straight 


line  AB  we  obtain  DE  =  AO  +  FE  =  AO  +  (BG)^  ,  and  hence 

a=w  +  (/-w)~,  (i) 

which  is  Launhardt's  formula. 

Moreover,  if  we  denote  by  max  L  the  greatest  load  on  the  en- 
tire piece,  and  by  min  L  the  least,  we  shall  have 

c_     min  L 
a     max  L' 
Hence 

min  L 


-  r, 

max  L 


(2) 


this  being  in  such  a  form  as  can  be  used.     Or  we  may  write  it 
thus: 

!/  —  u  min  L 
i+ j 

this  being  the  more  common  form. 

The  values  of  the  constants  as  determined  by  Wohler's  experi- 
ments, and  the  resulting  form  of  the  formula  for  Phcenix  axle- iron 
and  for  Krupp  cast-steel,  have  already  been  given  in  §  172. 


WEYRAUCH'S  FORMULA.  $2$ 

In  the  same  paragraph  are  given  the  corresponding  values  of 
by  the  safe  working-strength,  the  factor  of  safety  being  three. 

WEYRAUCH'S  FORMULA  FOR  ALTERNATE  TENSION  AND 

COMPRESSION. 

The  constants  used  in  this  formula  are  : 

i°.  u,  the  primitive  breaking-strength,  which  has  been  already 
defined. 

2°.  s,  the  vibration  breaking-strength  (Schwingungsfestigkeit) 
i.e.,  the  greatest  stress  per  unit  of  area,  of  which  the  piece  can 
bear,  without  breaking,  an  unlimited  number  of  applications, 
when  subjected  alternately  to  a  tensile,  and  to  a  compressive 
stress  of  the  same  magnitude. 

He  lets  a  =  greatest  stress  per  unit  of  area,  c=  greatest  stress 
of  the  opposite  kind  per  unit  of  area.  If  a  is  tension,  c  is  com- 
pression, and  vice  versa. 

Plot  the  values  of  -  as  abscissae,  and  those  of  a  as  ordinates, 
making  OA=u  (since  when  -  =  i,  a=w),  OC  =  i,  and  CB=s 

(since  when  —  =  i  ,  a  =s)  .    Then  will  any  curve 

which  passes  through  the  points  A  and  B 
have  for  its  ordinates  values  of  a  that  will 
satisfy  the  conditions  that  when  c=o,  a=u, 
and  when  c=s,  a=s. 

By  assuming  for  this  curve  the  straight  line  AB  we  obtain 

t  and  hence 


a=u-(u-s)-,  (4) 

which  is  the  Weyrauch  formula. 


526  APPLIED  MECHANICS. 

Moreover,  if  we  write 

c     max  U 
a     max  L  ' 

where  max  L=  greatest  load  on  the  piece,  and  max  Z/=  greatest 
load  of  opposite  kind,  so  that,  if  L  is  tension,  L'  shall  be  com- 
pression, and  vice  versa,  we  shall  have 

.max  L' 


this  being  in  a  form  suitable  to  use,  the  more  common  form  being 

(u—  s  max  L'  } 
i  -  ---  r  \  -  (6) 

u     max  L  J 

The  values  of  the  constants  as  determined  from  Wohler's 
experiments,  and  the  resulting  form  of  the  formulae  for  Phoenix 
axle-iron  and  for  Krupp  cast-steel,  are  given  in  §  176. 

GENERAL  REMARKS. 

In  each  case  the  value  of  a  given  by  the  formula  (3)  or  (6) 
is  the  breaking-strength  per  unit  of  area. 

If  either  of  these  values  of  a  be  divided  by  3,  we  have,  accord- 
ing to  Weyrauch,  the  safe  working-strength. 

WOHLER'S   EXPERIMENTAL   RESULTS. 

Wohler  himself  made  his  tests  upon  the  extremes  of  fibre 
stresses  of  which  a  piece  could  bear,  without  breaking,  an 
unlimited  number  of  applications.  He  gives,  as  a  summary  of 
these  results,  the  following:  — 

In  iron,  — 

Between  +16000  Ibs.  per  sq.  in.  and  —16000  Ibs.  per  sq.  in. 
+  30000       "          "        "  o  "          " 

+44000       "          "        "    +24000       "          " 

In  axle-steel,  — 

Between  +28000  Ibs.  per  sq.  in.  and  —28000  Ibs.  per  sq.  in. 
"       +48000       "  "        "  o  "          " 

"       +80000       "          "        "    +35000       "          " 


WOHLER'S  EXPERIMENTAL   RESULTS.  527 

In  untempered  spring  steel,  — 

Between  +50000  Ibs.  per  sq.  in.  and          o      Ibs.  per  sq.  in. 
«        -f  70000        "  "        "     +25000       "  " 

+  80000       "  "        "     +40000        "  " 

+  90000        "  "        "    +60000        "  " 

For  shearing  in  axle-steel,  — 

Between  +22000  Ibs.  per  sq.  in.  and  —22000  Ibs.  per  sq.  in. 
+  38000        "  "  o  " 

This  table  would  justify  the  use,  in  Launhardt's  and  Wey- 
rauch's  formulae,  of  the  following  values  of  u  and  s  ;  viz., — 
In  iron,  — 

u  =  30000  Ibs.  per  sq.  in., 

s  =  16000  Ibs.  per  sq.  in. 

In  axle  steel,  — 

u  =  48000  Ibs.  per  sq.  in., 
s  =  28000  Ibs.  per  sq.  in. 

In  untempered  spring  steel,  — 

u  =  50000  Ibs.  per  sq.  in. 

And  it  would  require,  that  if,  with  these  values  of  u,  and  the 
values  of  /  given  in  §§  172  and  176,  we  put 

c  —  24000 

in   Launhardt's  formula  for  iron,  we  ought  to  obtain  approxi- 
mately 

a  =  44000 ; 

and  if  we  put  c  =  35000  in  that  for  steel,  we  should  obtain 
approximately 

a  =  80000. 


528  APPLIED   MECHANICS. 


FACTOR   OF   SAFETY. 

We  have  seen  that  Weyrauch  recommends,  to  use  with 
Wohler's  results,  a  factor  of  safety  of  three  for  ordinary  bridge 
work  and  similar  constructions. 

Wohler  himself,  however,  in  his  "  Festigkeits  versuche  mit 
Eisen  und  Stahl,"  says, — 

i°.  That  we  must  guard  against  any  danger  of  putting  on 
the  piece  a  load  greater  than  it  is  calculated  to  resist,  by  assum- 
ing as  its  greatest  stress  the  actually  greatest  load  that  can 
ever  come  upon  the  piece ;  and 

2°.  This  being  done,  that  the  only  thing  to  be  provided  for 
is  the  lack  of  homogeneity  in  the  material. 

3°.  That  any  material  which  requires  a  factor  of  safety 
greater  than  two  is  unfit  for  use.  This  advice  would  hardly  be 
accepted  by  engineers,  however. 

He  also  claims  that  the  reason  why  it  is  safe  to  load  car- 
springs  so  much  above  their  limit  of  elasticity,  and  so  near 
their  breaking-load,  is,  that  the  variation  of  stress  to  which  they 
are  subjected  is  very  inconsiderable  compared  with  the  greatest 
stress  to  which  they  are  subjected. 

GENERAL   REMARKS. 

It  is  to  be  observed,  — 

i°.  'The  tests  were  all  made  on  a  good  quality  of  iron  and 
of  steel,  consequently  on  materials  that  have  a  good  degree  of 
homogeneity. 

2°.  The  specimens  were  all  small,  and  the  repetitions  of  load 
succeeded  each  other  very  rapidly,  no  time  being  given  for  the 
material  to  rest  between  them. 

3°.  No  observations  were  made  on  the  behavior  of  the  piece 
during  the  experiment  before  fracture. 


SHEARING-STRENGTH   OF  IRON  AND   STEEL.  529 

4°.  As  long  as  we  are  dealing  only  with  tension,  we  can  say 
without  error  that 

c_  _  min  L  t 
a       max  L ' 

but  as  soon  as  both  stresses  or  either  become  compression,  if 
the  piece  is  long  compared  with  its  diameter,  we  cannot  assert 
with  accuracy  the  above  relation,  nor  that 

c       max  U 
—  =  • 

a       maxZ 

and  hence  results  based  on  these  assumptions  must  be  to  a 
certain  extent  erroneous. 

5°.  When  a  piece  is  subjected  to  alternate  tension  and  com- 
pression, it  must  be  calculated  so  as  to  bear  either :  thus,  if 
sufficient  area  is  given  it  to  enable  it  to  bear  the  tension,  it  may 
not  be  able  to  bear  the  compression  unless  the  metal  is  'so  dis- 
tributed as  to  enable  it  to  withstand  the  bending  that  results 
from  its  action  as  a  column. 

While  Wohler's  tests  were  mostly  confined  to  ascertaining 
breaking-strengths,  the  later  experimenters  upon  this  subject, 
especially  Prof.  Bauschinger  at  Munich,  Mr.  Howard  at  the 
Watertown  Arsenal,  and  Prof.  Sondericker  at  the  Mass.  Institute 
of  Technology,  have  all  undertaken  to  study  the  elastic  change.3 
developed  in  the  material  by  repeated  stresses,  and  also,  to  some 
extent,  the  effect  upon  resistance  to  repeated  stress,  of  flaws,  of 
indentations,  and  of  sudden  changes  of  section,  including  sharp 
corners. 

They  all  agree  in  the  conclusion  that  flaws  and  indentations 
(even  though  very  slight)  and  sharp  corners,  including  keyways, 
reduce  the  resistance  to  repeated  stress  very  considerably. 

A  brief  account  will  be  given  of  some  of  their  principal  con- 
clusions. 


53°  APPLIED   MECHANICS. 


BAUSCHINGER'S    TESTS  ON  REPEATED  STRESSES. 

Bauschinger's  tests  upon  repeated  stress  include  work  upon 
the  properties  of  metals  at  or  near  the  elastic  limit.  Of  the 
properties  which  he  enumerates,  the  following  will  be  quoted 
here: 

(a)  The  sets  within  the  elastic  limit  are  very  small,  and  in- 
crease proportionally  to  the  load,  while  above  that  point  they 
increase  much  more  rapidly. 

(b)  With  repeated  loading,  inside  of  the  elastic  limit,  dropping 
to   zero    between    times,    we    find    each    time    the  same    total 
elongations. 

(c)  While   within    the    elastic    limit    the    elongations    remain 
constant  as  long  as  the  load  is  constant;   with  a  load  above  the 
elastic  limit  the  final  elongations  under  that  load  are  only  reached 
after  a  considerable  length  of  time. 

(d)  If  by  subjecting  a  rod  to  changing  stresses  between  an 
upper  and  lower  limit,  of  which  at  least  the  upper  is  above  the 
original  elastic  limit,  the  latter  were  either  unchanged  or  lowered, 
or  if,  in  the  case  of  its  being  raised,  it  were  to  remain  below  the 
upper  limit,  then  the  repetition  of  such  stresses  must  finally  end 
in  rupture,  for  each  new  application  of  the  stress  increases  the 
strain;   but  if    both    limits  of    the    changing    stress    are     and 
remain    below    the    elastic   limit,  the   repetition   will   not    cause 
breakage. 

(e)  Bauschinger  says  that  by  overstraining,  the  stretch  limit 
is  always  raised  up  to  the  load  with  which  the  stretching  was 
done;   but  in  the  time  of  rest  following  the  unloading  the  stretch 
limit  rises  farther,   so   that  it   becomes   greater   than   the  max- 
imum load  with  which  the  piece  was  stretched,  and  this  rising 
continues  for  days,  months,  and  years;    but,  on  the  other  hand, 
that   the   elastic   limit   is  lowered  by  the  overstraining,  often  to 
zero;    and    that    a   subsequent    rest    gradually  raises  it  until  it 
reaches,    after    several    days,    the    load    applied,    and    in    time 


EXPERIMENTS  WITH  A  REPEATED  TENSION  MACHINE.  531 

rises  above  this  ;  that,  as  a  rule,  the  modulus  of  elasticity 
is  also  lowered  under  the  same  circumstances,  and  is  also 
restored  by  rest,  and  rises  after  several  years  above  its 
original  magnitude. 

(/)  By  a  tensile  load  above  the  elastic  limit  the  elastic 
limit  for  compression  is  lowered,  and  vice  versa  for  a  compres- 
sive  load ;  and  a  comparatively  small  excess  over  the  elastic 
limit  for  one  kind  of  load  may  lower  that  for  the  opposite 
kind  down  to  zero  at  once.  Moreover,  an  elastic  limit  which 
has  been  lowered  in  this  way  is  not  materially  restored  by  a 
period  of  rest — at  any  rate,  of  three  or  four  days. 

(<£")  With  gradually  increasing  stresses,  changing  from 
tension  to  compression,  and  vice  versa,  the  first  lowering  of 
the  elastic  limit  occurs  when  the  stresses  exceed  the  original 
elastic  limit. 

(//)  If  the  elastic  limit  for  tension  or  compression  has  been 
lowered  by  an  excessive  load  of  the  opposite  kind,  i.e.,  one  ex- 
ceeding the  original  elastic  limit,  then,  by  gradually  increasing 
stresses,  changing  between  tension  and  compression,  it  can 
again  be  raised,  but  only  up  to  a  limit  which  lies  considerably 
below  the  original  elastic  limit. 

EXPERIMENTS   WITH   A  REPEATED   TENSION   MACHINE. 

Bauschinger  states  that  in  1881  he  acquired  a  machine 
similar  to  that  used  by  Wohler  for  repeated  application  of  a 
tensile  stress. 

The  plan  of  the  experiments  which  he  made  with  it,  and 
which  are  detailed  in  the  I3th  Heft  of  the  Mittheilungen,  is  as 
follows : 

From  a  large  piece  of  the  material  there  were  cut  at  least 
four,  and  sometimes  more,  test-pieces  for  the  Wohler  machine. 
One  of  them  was  tested  in  the  Werder  machine  to  determine 
its  limit  of  elasticity  and  its  tensile  strength  ;  the  others  were 


532  APPLIED   MECHANICS. 

tested  in  the  Wohler  machine,  so  arranged  that  the  upper  limit 
of  the  repeated  stress  should  be,  for  the  first  specimen,  near 
the  elastic  limit  ;  for  the  second,  somewhat  higher,  etc.,  the 
lower  limit  being  in  all  cases  zero. 

From  time  to  time  the  test-pieces,  after  they  had  been  sub- 
jected to  some  hundred  thousands,  or  some  millions,  of  repeti- 
tions, were  taken  from  the  Wohler  machine  and  had  their  limits 
of  elasticity  determined  in  the  Werder  machine. 

The  tables  of  the  tests  are  to  be  found  in  the  Mittheilungen, 
and  from  them  Bauschinger  draws  the  following  conclusions : 

i°.  With  repeated  tensile  stresses,  whose  lower  limit  was 
zero,  and  whose  upper  limit  was  near  the  original  elastic  limit, 
breakage  did  not  occur  with  from  5  to  16  millions  of  repeti- 
tions. 

Bauschinger  says  that  in  applying  this  law  to  practical  cases 
we  must  bear  in  mind  two  things  :  (a)  that  it  does  not  apply 
when  there  are  flaws,  as  several  specimens  which  contained  flaws, 
many  of  them  so  small  as  to  be  hardly  discoverable,  broke  with 
a  much  smaller  number  of  repetitions ;  (b)  another  caution  is 
that  we  should  make  sure  that  we  know  what  is  really  the  origi- 
nal elastic  limit,  as  this  varies  very  much  with  the  previous 
treatment  of  the  piece,  especially  the  treatment  it  received 
during  its  manufacture,  and  it  may  be  very  small,  or  it  may  be 
very  near  the  breaking-strength. 

2°.  With  oft-repeated  stresses,  varying  between  zero  and  an 
upper  stress,  which  is  in  the  neighborhood  of  or  above  the 
original  elastic  limit,  the  latter  is  raised  even  above,  often  far 
above,  the  upper  limit  of  stresses,  and  the  higher  the  greater 
the  number  of  repetitions,  without,  however,  its  being  able  to 
exceed  a  known  limiting  value. 

3°.  Repeated  stresses  between  zero  and  an  upper  limit, 
which  is  below  the  limiting  value  of  stress  which  it  is  possible 
for  the  elastic  limit  to  reach,  do  not  cause  rupture  ;  but  if  the 
upper  limit  lies  above  this  limiting  value,  breakage  must  occur 
.after  a  limited  number  of  repetitions. 


EXPERIMENTS  WITH  A  REPEATED  TENSION-MACHINE.  533 

4°.  The  tensile  strength  is  not  diminished  with  a  million 
repetitions,  but  rather  increased,  when  the  test-piece  after  hav- 
ing been  subjected  to  repeated  stresses  is  broken  with  a  steady 
load. 

5°.  He  discusses  here  the  probability  of  the  time  of  forma- 
tion of  what  he  considers  to  be  a  change  in  the  structure  of 
the  metal  at  the  place  of  the  fracture. 

Besides  the  above  will  be  given  the  numerical  values  which 
Bauschinger  obtained  for  carrying  strength  and  for  primitive 
safe  strength  as  average  values. 

i°.  For  wrought-iron  plates  : 

/  =  49500  Ibs.  per  sq.  in. 
u  —  28450    "      "     "     " 

2°.  For  mild-steel  plates  (Bessemer) : 

t  =  62010  Ibs.  per  sq.  in. 

«=  34140    "      "     "     " 

>     i 

3°.  For  bar  wrought-iron,  80  mm.  by  10  mm. : 

t  =  57600  Ibs.  per  sq.  in. 
u=  31290    "      "     "     " 

4°.  For  bar  wrought-iron,  40  mm.  by  10  mm. : 

/  ==  57180  Ibs.  per  sq.  in. 
u  =  34140    "      "     "     " 

5°.  For  Thomas-steel  axle  : 

t  =  87050  Ibs.  per  sq.  in. 
u  =  42670    "      "     "     " 

6°.  For  Thomas-steel  rails  : 

/  =  84490  Ibs.  per  sq.  in. 
u  =  39820    "      "     "     " 


534  APPLIED    MECHANICS. 

7°.  For  Thomas-steel  boiler-plate : 

£=57600  Ibs.  per  sq.  in. 
u  =  34140    "      "     "     " 

For  Thomas-steel  axle,  and  Thomas-steel  rails,  Bauschinger's 
obtained  for  the  vibration  breaking-strength  the  same  values  as 
those  for  primitive  breaking-strength.  His  experiments  on  the 
other  five  niaterials,  however,  give  lower  values  for  5  than  for 
u.  These  values  will  not  be  quoted  here,  however,  because  they 
were  obtained  from  experiments  upon  rotating  bars  of  rectangular 
section  transversely  loaded. 

EXPERIMENTS  UPON  ROTATING  SHAFTING  SUBJECTED  TO  TRANS- 
VERSE  LOADS,    BY   PROF.    SONDERICKER. 

Accounts  of  these  tests  are  to  be  found  in  the  Technology 
Quarterly  of  April,  1892,  and  of  March,  1899.  In  every  case  the 
(transverse)  loads  were  so  applied,  that  a  certain  portion,  greater 
than  ten  inches  in  length,  was  subjected  to  a  uniform  bending- 
moment.  At  various  times,  the  shaft  was  stopped,  the  load  was 
removed,  then  replaced,  and  again  removed,  and  measurements 
made  of  the  strains  and  sets.  The  diameter  of  the  shaft  was, 
in  every  case,  approximately  one  inch.  Some  extracts  from  the 
paper  of  March,  1899,  will  be  given.  The  investigations  were 
conducted  along  two  lines. 

i°.  The  determination  of  elastic  changes,  resulting  from 
the  repeated  stresses,  and  the  influence  of  such  changes  in  pro- 
ducing fracture. 

2°.  The  influence  of  form,  flaws,  and  local  conditions  generally 
in  causing  fracture. 

Accurate  measurements  of  the  elastic  strains,  and  sets  were 
made  at  intervals  during  each  test.  Characteristic  curves  of  set 
indicate  the  general  character  of  the 
changes  which  occurred  in  the  set,  the 
abscissa?  being  the  number  of  revo- 
lutions, and  the  ordinates  the  amount 
of  the  set,  a  is  the  characteristic  curve 


EXPERIMENTS  WITH  A  REPEATED  TENSION  MACHINE.    535 


for  wrought-iron,  and  also  occurred  in  one  kind  of  soft  steel.  No 
change  is  produced  until  the  elastic  limit  is  reached,  and  then 
the  change  consists  in  a  decrease  of  set.  b  is  the  characteristic 
curve  for  all  the  steels  tested  with  the  single  exception  mentioned. 
It  is  the  reverse  of  the  preceding,  beginning  commonly  below  the 
elastic  limit,  and  consisting  of  an  increase  of  set;  rapid  at  first, 
but  finally  ceasing.  Under  heavy  loads,  the  increase  of  set  L 
very  rapid,  and  ceases  comparatively  quickly.  Accompanying 
the  change  of  set  there  is  a  change  in  the  elastic  strain  in  the 
same  direction  but  much  smaller  in  amount.  From  the  fact  that 
these  changes  finally  cease,  we  conclude  that,  if  of  sufficiently 
small  magnitude,  they  do  not  necessarily  result  in  fracture. 

The  table  on  page  536  gives  a  number  of  his  results. 

Regarding  these  results  he  says : 

i°.  In  several  cases,  changes  would  have  been  detected  under 
smaller  stresses  had  observations  been  taken. 

2°.  Changes  of  set  may  be  expected  to  begin  at  stresses  vary- 
ing from  J  to  J  of  the  tensile  strength. 

3°.  The  set  does  not  appear  to  have  a  notable  influence  in 
causing  fracture  until  it  reaches  o".ooi  or  o".oo2  in  a  length  of 
ten  inches. 

4°.  The  effect  of  rest  is  to  decrease  the  amount  of  set.  In 
most  cases,  however,  the  set  lost  is  soon  regained,  when  the  bar  is 
again  subjected  to  repeated  stress,  especially  in  the  case  of  the 
harder  steels. 

Prof.  Sondericker  also  cites  a  few  experiments  to  determine 
the  loss  of  strength  due  to  indentations,  grooves,  and  key  ways. 
In  one  case,  the  result  of  cutting  a  groove  around  the  steel  shaft 
about  o".cx>3  deep  was  a  loss  of  strength  of  about  40  per  cent, 
while  similar  results  were  obtained  with  indentations,  and  with 
square  shoulders.  He  also  cites  the  case  of  two  pieces  of  steel 
shafting  united  by  a  coupling,  where  the  result  of  cutting  the 
necessary  keyways  in  the  shafts  caused,  apparently,  a  loss  of 
about  50  per  cent. 


536 


APPLIED    MECHANICS. 


Tensile  Prop- 

i 

!  erties  of  the 

Revolu- 

Metal. 

tions  at 

Maxi- 

X 

Material. 

Elastic 
Limit 

Tensile 
St'gth 

Stress 
per 
Sq.  In. 

which 
Change 
was  First 
Ob- 

Revolu- 
tions. 

mum 
Observed 
Sets. 

Remarks. 

rt 

B 

per 
Sq.  In. 
Lbs. 

per 
Sq.  In. 
Lbs. 

Lbs. 

served. 

Inches. 

D 

Wt.-Iron 

15700 

45080 

30000 

42300 

86400 

<  .01200 

Broke  at  one  end  at 

shoulder,  and  at 

other  where  arm 

was  attached. 

40 

1  • 

24000 

50700 

24000 

1500000 

1500000 

26000 

2427000 

Broke. 

i 

'  • 

25900 

51390 

26000 

2214000 

2285000 

.00026 

Broke  near  center. 

2 

" 

25900 

5139° 

32000 

486000 

486000 

.00136 

Broke  at  mark  burn- 

. 

ed  by  electric  cur- 

rent. 

3 

1  ' 

23400 

50510 

24000 

6593000 

6593000 

.00037 

24000 

4059000 

.  OOOI  I 

. 

25000 

8962000 

.00016 

26000 

3932000 

.00022 

27000 

8155000 

.00037 

28000 

589000 

.00038 

Broke  at  shoulder. 

4 
33 

Steel 

23400 
24800 

50510 
47400 

28000 
32000 

2506000 
85900 

2506000 
89750 

.00042 

.00771 

Broke  at  center. 
Broke  outside  of  arm 

near  bearing;  color 

blue  black. 

34 

•  • 

24800  47400 

32000 

103500 

116600 

.00832 

Do. 

21 

'  ' 

30400  62590 

32000 

4395000 

4395000 

.  OOO29 

34000 

8339000 

.00032 

36000 

4627000 

.00041 

36000 

1428000 

.OO008 

After  resting  unload- 

ed 1  8  days. 

38000 

3769000 

.00023 

40000 

4523000 

.00054 

42000 

505000 

.00072 

Broke  near  shoulder. 

54 

•  • 

42000 

63130 

45000 

163000 

163000 

.00312 

Broke  at  shoulder. 

50 

4  • 

23200 

73760 

30000 

339000 

339000 

.OOIOO 

35000 

16400 

.00282 

Not  broken. 

25 

'  ' 

38300 

78010 

40000 

5031000 

5031000 

.00028 

42000 

2483000 

.00046 

Broke  at  shoulder. 

26 

" 

38300 

78010 

40000 

20838000 

20838000 

.00037 

42000 

3311000 

.00044 

Not  broken. 

18 

" 

50000 

•  81010 

36000 

6463000 

6982000 

.00052 

Broke  where  arm  was 

attached. 

iQ 

1  • 

50000 

81010 

36000 

7252000 

7686000 

.00069 

Broke  at  shoulder. 

20 

•  ' 

50000 

81010 

34000 

2I22IOOO 

21  22IOOO 

.00028 

36000 

13577000 

.00067 

38000 

2263OOO 

.00113 

38000 

9237000 

.00116 

After  resting  6  mos. 

unloaded. 

40000 

932000 

.00177 

Broke  at  shoulder. 

S3 

«• 

58000 

96580 

50000 

24000 

146500 

.00249 

Broke  at  shoulder. 

58 

58000 

96580 

45000 
50000 

50100 

50100 
156900 

.00020 
.00289 

Broke  near  middle. 

29 

" 

54000 

104480 

40000 

5257000 

5257000 

.00046 

42000 

7125000 

.00067 

44000 

4626000 

.00100 

46000 

6/60000 

.00145 

48000 

4965000 

.00196 

50000 
50000 

I  7OOOO 
1000 

.00197 
.00203 

After  24  days  rest  un- 

loaded ;  not  broken. 

55 

" 

50000 

104830 

35000 

276900 

276900 

.00060 

40000 

237900 

.00274 

50000 

22530 

.00615 

Broke  near  shoulder; 

color  dark  straw. 

57 

50000 

104830 

60000 

14300 

14900 

.00768 

Broke  near  shoulder; 
color  dark  blue. 

EXPERIMENTS  WITH  A  REPEATED    TENSION  MACHINE.  537 

TESTS     OF     ROTATING     SHAFTING     UNDER    TRANSVERSE     LOAD,     BY 
MR.    HOWARD    AT   THE   WATERTOWN   ARSENAL. 

A  large  number  of  tests  of  this  character  have  been  made  at 
the  Watertown  Arsenal.  A  few  extracts  will  be  given  from  the 
remarks  of  Mr.  Howard  upon  the  subject,  which  may  be  found 
in  the  Technology  Quarterly  of  March,  1899,  as  follows: 

"In  the  Watertown  tests,  two  principal  objects  have  been 
in  view,  namely,  to  ascertain  the  total  number  of  repetitions  of 
stresses  necessary  to  cause  rupture,  and  to  observe  through  what 
phases  the  physical  properties  of  the  metal  pass  prior  to  the 
limit  of  ultimate  endurance.  The  Watertown  tests  have  included 
cast-iron,  wrought-iron,  hot  and  cold  rolled  metal,  and  steels 
ranging  in  carbon  from  o.i  per  cent  to  i.i  per  cent,  also  milled 
steels.  The  fibre-stresses  have  ranged  from  10000  pounds 
per  square  inch  on  the  cast-iron  bars  up  to  60000  pounds  pel- 
square  inch  on  the  higher  tensile-strength  steel  bars. 

The  speed  of  rotation  was  from  400  per  minute  up  to  2200 
per  minute,  in  different  experiments.  Observations  were  made 
on  the  deflection  of  the  shafts,  and  on  the  sets  developed.  It 
was  early  observed  that  intervals  of  rest  were  followed  by  tem- 
porary reduction  in  the  magnitude  of  the  sets.  In  the  Report 
of  Tests  of  Metals  of  1888,  he  says  the  deflections  tend  to 
diminish  under  high  speeds  of  rotation,  when  the  loads  exceed 
the  elastic  limit  of  the  metal,  and  tend  to  cause  permanent  sets; 
but,  on  the  other  hand,  when  the  elastic  limit  is  not  passed,  the 
deflections  are  the  same  within  the  range  of  speeds  yet  experi- 
mented upon. 

Efforts  were  inaugurated  at  this  time  to  ascertain  the  effect 
of  repeated  alternate  stresses  on  the  tensile  properties  of  the 
metal,  and  it  appeared  that  such  treatment  tended  to  raise  the 
tensile  strength  of  the  metal  before  rupture  ensued. 

Concerning  the  limit  of  indefinite  endurance  to  repeated 
stress  we  know  but  very  little.  In  most  experiments  rupture 
occurs  after  a  few  thousand  repetitions,  so  high  have  been  the 


533 


APPLIED  MECHANICS. 


applied  stresses.  Examples  are  not  uncommon  in  railway  prac- 
tice of  axles  having  made  200000000  rotations.  In  order  to 
establish  a  practical  limit  of  endurance,  indefinite  endurance, 
if  we  choose  to  call  it  so,  our  experimental  stresses  will  need  to 
be  somewhat  lowered,  or  new  grades  of  metal  found. 

The  following  table  which  accompanied  the  Watertown 
Arsenal  Exhibit  at  the  Louisiana  Purchase  Exposition  gives  a 
summary  of  some  of  the  repeated  stress  tests  upon  three  different 
grades  of  steel: 

STEEL  BARS. 
Tensile  Tests  and  Repeated  Stress  Tests  on  Different  Carbon  Steels. 


Tensile  Tests. 

Repeated  Stress  Tests. 

Mechan- 

Maxi- 

Elastic 

Ten- 

Elon- 

Con- 

ical 

mum 

Mechan- 

Description. 

Limit 
per 

sile 
St'gth 

gation 
in  4 

trac- 
tion of 

Work  at 
Rupture 

Fiber 
Stress 

Number 
of  Rota- 

ical Work 
at  Rup- 

Sq. In. 
Lbs. 

per 

Ins. 
Per  ct. 

Area 
Per  ct. 

per 
Cu.In. 

per 
Sq.  In. 

tions  at 
Rupture. 

ture  per 
Cu.  In. 

Lbs. 

Ft.-lbs. 

Lbs. 

Ft.-lbs. 

(60000 

6470 

32835 

50000 

17790 

62635 

0.17  Carbon  steel  . 

51000 

68000 

33-5 

51-9 

982 

45000 
j  40000 

70400 
293500 

201960 
665290 

1  35000 

5757920 

9992390 

30000 

*236ooooo 

*295ooooo 

60000 

12490 

63387 

50000 

93160 

328000 

0.5  5  Carbon  steel. 

57ooo 

106100 

16.2 

18.7 

1.047 

45000 
40000 

166240 
455350 

476900 
1032130 

35000 

9007  20 

1563125 

30000 

*i  9870000 

*24838ooo 

60000 

37250 

189044 

55000 

93790 

399780 

0.82  Carbon  steel. 

63000 

142250 

8.5 

6.5 

888 

50000 
45000 

213150 
605460 

750465 
1736910 

40000 

*i756oooo  *409730oo 

35000 

*I9220000 

*33635ooo 

*  Not  ruptured. 
GENERAL   REMARKS. 

That  the  amount  of  detailed  information  regarding  repeated 
stresses  is  small  compared  with  what  is  needed  will  be  evident 
when  we  consider  the  number  of  cases  in  which  metal  is  subjected 
to  such  stresses  in  practice,  among  which  are  shafting,  connecting- 
rods,  parallel  rods,  propeller-shafts,  crank-shafts,  railway  axles, 
rails,  riveted  and  other  bridge  members,  etc.  In  the  case  of 


TORSIONAL  STRENGTH  OF  WROUGHT  IRON  AND  STE^L.    539 


some  of  them,  notably,  railway  axles,  attempts  have  been  made 
to  base  specifications  for  the  material  upon  such  tests  as  have 
become  available  upon  repeated  stresses. 

§  229.  Shearing-strength  of  Iron  and  Steel. — Some  of 
the  most  common  cases  where  the  shearing  resistance  of  iron 
and  steel  is  brought  into  play  are  : 

i°.  In  the  case  of  a  torsional  stress,  as  in  shafting. 

2°.  In  the  case  of  pins,  as  in  bridge-pins,  crank-pins,  etc. 

3°.  In  the  case  of  riveted  joints. 

The  so-called  apparent  outside  fibre-stress  at  fracture,  as 
determined  from  experiments  on  torsional  strength,  is  found 
to  be  not  far  from  the  tensile  strength  of  the  metal,  and  is,  of 
course,  greater  than  the  shearing-strength,  for  the  same  reasons  as 
render  the  modulus  of  rupture  greater  than  the  actual  outside 
fibre-stress  at  fracture  in  transverse  tests. 

Moreover,  the  shearing  strength  of  wrought-iron  rivets  is 
shown  by  experiment  to  be  about  f  the  tensile  strength  of  the 
rivet  metal. 

In  regard  to  cast-iron,  Bindon  Stoney  found  the  shearing  and 
tensile  strength  about  equal. 

The  cases  where  shearing  comes  in  play  in  wrought-iron  and 
steel  will  therefore  be  treated  separately. 

§  230.  Torsional  Strength  of  Wrought-iron  and  Steel. — 
The  method  formerly  followed,  and  in  use  by  some  at  the  present 
day,  was  to  compute  the  strength  of  a  shaft  from  the  twisting- 
moment  only,  neglecting  the  bending,  but  varying  the  working- 
strength  per  square  inch  to  be  used  according  to  the  character 
of  the  service.  It  is  generally  the  fact,  however,  that  when 
shafting  is  running  the  pulls  of  the  belts  create  a  bending  back- 
wards and  forwards,  bringing  the  same  fibre  alternately  into 
tension  and  compression;  and  this  is  combined  with  the  shearing- 
stresses  developed  due  to  the  twisting-moment  alone.  At  the 
two  extremes  of  these  general  cases  are : 

i°.  The  case  when  the  portion  of  a  shaft  between  two  hangers 


540  APPLIED    MECHANICS. 

has  no  pulleys  upon  it,  and  when  the  pulls  on  the  neighboring 
spans  are  not  so  great  as  to  deflect  this  span  appreciably.  That 
is  a  case  of  pure  torsion:  and  if  the  shaft  is  running  smoothly, 
with  no  jars  or  shocks,  and  no  liability  to  have  a  greater  load 
thrown  upon  it  temporarily,  we  may  compute  it  by  the  usual 
torsion  formula,  given  in  §  212;  using  for  breaking-strength  of 
wrought-iron  and  steel  the  so-called  apparent  outside  fibre-stress 
at  fracture  as  determined  from  torsional  tests,  and  a  factor  of 
safety  six,  and  such  a  proceeding  will  probably  give  us  a  reasonable 
degree  of  safety. 

2°.  The  case  when,  pulleys  being  placed  otherwise  than  near 
the  hangers,  the  belt-pulls  are  so  great  that  the  torsion  becomes 
insignificant  compared  with  the  bending,  and  then  it  would  be 
proper  to  compute  our  shaft  so  as  not  to  deflect  more  than  y^-g-g- 
of  its  span  under  the  load,  or  better,  not  more  than  yeV  o :  °f 
course  we-  should  compute  also  the  breaking  transverse  load,  and 
see  that  we  have  a  good  margin  of  safety. 

In  other  cases,  the  methods  pursued,  the  first  two  of  which 
are  incorrect,  have  been 

i°.  By  using  the  ordinary  torsion  formula  combined  with  a 
large  factor  of  safety. 

2°.  By  computing  the  shaft  also  for  deflection,  and  providing 
that  its  deflection  shall  not  exceed  rsV<r  or  Trinr  °f  its  span. 

This,  however,  neglects  the  torsion,  and  also  the  rapid  change 
of  stress  upon  each  fibre  from  tension  to  compression. 

3°.  By  using  the  formula  of  Grashof  or  of  Rankine  for  com: 
bined  bending  and  twisting,  with  the  constants  that  have  been 
derived  from  experiments  on  simple  tension  or  simple  torsion. 

The  results  given  on  pages  544  and  545  are  from  pieces  of 
shafting  of  considerable  length.  As  has  been  stated,  the  so-called 
"apparent  outside  fibre-stress  at  fracture"  appears  to  be  not  very 
far  from  the  tensile  strength  of  the  material,  and  the  torsional 
modulus  of  elasticity  appears  to  be  from  three-eighths  to  two- 
fifths  of  the  tensile  modulus  of  elasticity. 


TORSIONAL  STRENGTH  OF  WROUGHT-IRON  AND  STEEL.  541 

Under  certain  circumstances  the  bending  may  have  the 
greatest  influence,  while  the  twisting  may  be  predominant  in 
others,  or  their  influence  may  be  equally  divided.  Which  of  these 
is  the  case  will  depend  upon  the  location  of  the  hangers  and  of 
the  pulleys,  the  width  of  the  belts,  etc.,  etc. 

As  to  the  formulae  which  take  into  account  both  twisting  and 
bending,  there  are  two,  both  of  which  are  based  upon  the  theory 
of  elasticity.  The  first,  which  is  the  most  correct  from  a  theo- 
retical point  of  view,  is  that  given  by  Grashof  and  other  writers 
on  the  theory  of  elasticity,  and  is 


where  Mi='greatest  bending-moment  ; 
M%  =  greatest  twisting-moment  ; 
r  =  external  radius  of  shaft; 

/  =  moment  of  inertia  of  section  about  a  diameter; 
/  =  greatest  allowable  stress  at  outside  fibre; 
w  =  a  constant  depending  on  the  nature  of  the  material. 
In  the  case  of  iron  or  steel  the  value  of  m  is  often  taken  as  4, 
though  it  is,  in  most  cases,  nearer  3.     When  m  =  4  we  have 


f__r 

'-/ 

The  other  formula,  which  is  also  based  upon  the  theory  of 
elasticity,  but  which  is  not  as  correct,  is  that  given  by  Rankine, 
and  is 


With  a  view  to  determine  the  behavior  of  shafting  under  a 
combination  of  twisting  and  bending,  suitable  machinery  was 
erected  in  the  engineering  laboratories  of  the  Mass.  Institute  of 
Technology,  and  a  number  of  tests  were  made. 


542 


APPLIED    MECHANICS. 


The  principal  points  of  the  method  of  procedure  are  the 
following,  viz.: 

i st.  The  shaft  under  test  is  in  motion,  and  is  actually  driving 
an  amount  of  power  which  is  weighed  on  a  Prony  brake. 

2d.  A  tr  nsverse  load  is  applied  which  may  be  varied  at  the 
option  of  the  experimenter,  and  which  is  weighed  on  a  platform 
scale. 

3d.  The  proportion  between  the  torsional  and  transverse 
loads  may  be  adjusted  to  correspond  with  the  proportion  be- 
tween the  power  transmitted  and  the  belt-pull  sustained  by  a 
shaft  in  actual  use. 

4th.  Tests  are  made  not  only  of  breaking-strength,  but  also 
angle  of  twist  and  deflection  under  moderate  loads  are  measured. 

The  following  table  will  give  the  results  oT  the  tests  on 
iron  shafts,  and  they  will  then  be  discussed  : 


Time 

•ji/i, 

*., 

A, 

/a, 

No. 

of 

Total 

H.  P. 

max. 

max. 

max. 

max. 

j- 

of 
Test. 

run- 
ning, 
min- 

revolu- 
tions. 

trans- 
mitted. 

bending 
moment. 

twisting 
moment. 

bend, 
fibre 

twist, 
fibre 

Grashof. 

Ran- 
kine. 

Diam. 
ins. 

utes. 

In.-lbs. 

In.-lbs. 

stress. 

stress. 

8 

37-5 

7040 

11.717 

11514.1 

3926.4 

60024 

10234 

62162 

6i755 

"•25 

9 

200 

38839 

8.181 

10507.8 

2656.8 

54777 

6925 

55876 

55671 

".25 

10 

l62 

31641 

5.291 

9891.0 

1714.6 

5*562 

4469 

52062 

5*976 

".25 

ii 

553 

108002 

4-331 

9241.7 

1399.2 

48179 

3647 

48539 

48769 

"•25 

12 

408 

80694 

6.276 

9241.7 

2027.6 

48179 

5287. 

48911 

48769 

".25 

13 

98 

19333 

6.342 

8917.1 

2028.2 

46485 

5287 

47245 

47105 

"•25 

14 

423 

82741 

6.283 

8917.1 

2029.7 

46485 

5290 

47246 

47106 

"•25 

15 

565 

108739 

6.192 

8592-5 

2031.6 

44793 

5295 

45582 

45436 

"•25 

10 

443 

88208 

6.338 

8267.8 

2026.8 

43100 

5283 

439H 

437*3 

".25 

17 

95i 

185233 

6.283 

3781-5 

2029.7 

38503 

10333 

41768 

41117 

" 

14  874 

8218 

84185 

0X028 

68 

It 

2O 

*T"°JT 

7,  562 

7976 

2394 

82112 

I2l88 

8  o  i 

„ 

21 

9.972 

/y/^ 
8917 

3232 

90793 

l6454 

03716 

„ 

22 

T  C      T  CQ 

8017 

2468l 

O86l2 

„ 

•*•->•  A  jy 
2  .  955 

°yL  / 

7652 

945 

77OI7 

48ll 

82 

II 

//y1  j 

T-OI>  L 

TOR  SIGNAL    STRENGTH  OF   WROUGHT-IRON,  ETC.      543 

In  19  to  23  inclusive  the  number  of  revolutions  was  small  and 
the  outside  fibre  stress  at  fracture  was  correspondingly  large. 

Two  specimens  of  the  \'  '.25  shafting  and  two  of  the  i" 
were  tested  for  tension,  the  results  being  as  follows  : 

Breaking-strength,  per  sq.  in. 

,,         ,.  (  No.  i    •          .  46800 

,".35  diameter  ]  NQ  2    ...... 

Average      ... 


Average     ....     60250 
As  to  conclusions  : 

1st.  It  is  plain  from  these  results  that  a  shaft  whose  size  is 
•determined  by  means  of  the  results  of  a  quick  test  would  be 
too  weak,  and  that  our  constants  should  be  obtained  from  tests 
which  last  for  a  considerable  length  of  time. 

2d.  A  perusal  of  the  tables  will  show  that  the  results  ob- 
tained apply  more  to  the  bending  than  to  the  twisting  of  a 
shaft,  as  the  transverse  load  used  in  these  tests  was  so  large 
compared  with  the  twist  as  to  exert  the  controlling  influence. 
This  will  be  plain  by  a  comparison  of  the  values  of  fltft, 
and/. 

3d.  Nevertheless,  the  bending-moments  actually  used  were 
generally  less  than  such  as  might  easily  be  realized  in  practice 
with  the  twisting-moments  used. 

4th.  It  seems  fair  to  conclude  that,  in  the  greater  part  of 
cases  where  shafting  is  used  to  transmit  power,  as  in  line-shaft- 
ing or  in  most  cases  of  head  shafting,  the  breaking  is  even  'more 
liable  to  occur  from  bending  back  and  forth  than  from  twist- 
ing, and  hence  'that  in  no  such  case  ought  we  to  omit  to 
make  a  computation  for  the  bending  of  the  shaft  as  well  as  the 
twist. 

5th.  As  to  the  precise  value  of  the  greatest  allowable  out- 
side fibre  stress  to  be  used  in  the  Grashof  formula,  it  is  plain 


544 


APPLIED    MECHANICS. 


that  it  is  not  correct  to  use  a  value  as  great  as  the  tensile 
strength  of  the  iron,  and  while  the  tests  show  that  this  figure 
should  not  for  common  iron  exceed  40000  Ibs.  per  square  inch, 
it  is  probable  that  tests  where  a  longer  time  is  allowed  for 
fracture  will  show  a  smaller  result  yet. 

TORSIONAL   TESTS   OF  WROUGHT-IRON. 


Norway  Iron. 

Burden's  Best. 

x 

0 

C  " 

-*-» 

^   c 

(0    O 

J=  <«' 

--. 

be 

C  A 

*-> 
M  rt 

0 

01    '"•' 

V 

OJ  _C 

(fl  C 

is, 

3    C 
"3  .'". 

ir.  o 

8* 

§! 

g& 

."S  -E 

2  -E 

Diameter.  (In< 

Distance  betw 
Grips.  (Incl 

Maximum  Twi 
Moment.  (I 
Lbs.) 

Number  of  Tu 
between  Gri 
Fracture. 

sarent  Out 
ibre  Stress, 
-bs.  per  sq. 

•a  >,cr 
o.ti  tn 

Diameter  of  C 
section.  (In< 

|£ 

u 

p 

MaximumTwi 
Moment.  (] 
Lbs.) 

Number  of  Tu 
between  Gri 
Fracture. 

Apparent  Out 
Fibre  Stress 
(Lbs.  per  sq. 

Shearing  Modi 
of  Elasticity 
(Lbs.  per  sq 

au^ 

j=  °^ 
1/3 

.00 

70.40 

72360 

16.50 

46065 

11406000 

.01 

63.8 

85050 

9-50 

533oo 

11300000 

.02 

72.00 

74970 

16.00 

46600 

13215000 

.01 

59-0 

86400 

8.63 

54200 

11500000 

•°3 

71-3° 

72000 

14.00 

43757 

12902000 

.01 

53-o 

84510 

6.87 

53000 

11200000 

02 

70.40 

74520 

17.00 

46321 

12247000 

.00 

58-8 

87480 

8.4o 

557oo 

II600000 

.02 

69.80 

72000 

14-25 

45837 

12738000 

.00 

65-5 

85410 

8.52 

544oo 

11600000 

03 

70.30 

74880 

15-50 

456Qa 

11361000 

.01 

60.2 

8559° 

8.82 

537°° 

1  1200000 

•03 

74880 

20.00 

44658 

11957000 

.02 

58.5 

85140 

8.05 

52600 

II3OOOOO 

03 

70.20 

7956o 

16.00 

48437 

11554000 

.OO 

57-0 

82650 

7-31 

52600 

II500000 

•03 

84 

74880 

15-5 

45590 

11900000 

.02 

57-8 

86580 

8.54 

535oo 

II2OOOOO 

•54 

54 

35100 

12.  0 

48950 

9840000 

.02 

59-5 

86040 

8.61 

53200 

II200000 

52 

49 

34200 

11.25 

49600 

11410000 

.02 

60.0 

87840 

8-93 

543°° 

11300000 

•53 

53 

33840 

8.50 

48120 

Il6oOOOO 

.OX 

60.0 

88200 

8.48 

544oo 

I  1200000 

•3-384.0 

II  .  IO 

48110 

.OI 

53-  3 

87480 

7.85 

1  I  300000 

•53 

49 

6  J°T-U 

34920 

14.56 

49650 

11840000 

.OI 

59-5 

83970 

8.01 

52700 

11400000 

•52 

53 

34200 

8.98 

49600 

12480000 

.01 

59-5 

84780 

8.32 

53200 

II200000 

•25 

70 

111960 

5-8o 

50060 

11830000 

•03 

61.0 

83520 

8.98 

50900 

I  I  IOOOOO 

•27 

75 

106920 

12-30 

46600 

10900000 

.00 

63.0 

84050 

9.24 

535°o 

11700000) 

•25 

70 

108360 

10.90 

48500 

IlSooooo 

.02 

60.3 

85950 

7-94 

53*00 

1I2OOOOO 

•25 

76 

109800 

9.90 

49100 

11700000 

.OI 

60.0 

84600 

8.62 

53100 

11400000 

•23 

70 

113670 

11.00 

52200 

12000000 

.02 

61.0 

83520 

8.50 

51600 

IIOOOOOO 

.26 

7° 

107640 

IO.OO 

47500 

II400000 

.OO 

60.5 

86040 

8.80 

54000 

11600000! 

Refined  Iron. 

.01 
2.01 

3o!8 

85680 
87480 

siii 

53700 
54900 

11300000 
11500000 

.03 

72.10 

84240 

4.00 

51285 

I247IOOO 

2.01 

47-8 

85860 

7.24 

539°o 

11300000 

.03 

71  .20 

66960 

40646 

12576000 

2.OI 

59-4 

85050 

8.92 

533°° 

11500000 

•03 

70.50 

70200 

3-50 

41743 

II372000 

2.01 

60.9 

86400 

8.75 

54200 

11300900 

72.10 

72000 

2.30 

43834 

10960000 

2.00 

59-5 

86400 

9.08 

55°oo 

11700000 

•03 

71.80 

61920 

2.80 

36820 

11393000 

2.01 

58.5 

85650 

8.30 

537°° 

11500000 

•03 

7!-3° 

68760 

2.50 

40887 

H36OOOO 

2.00 

58.5 

84870 

7-87 

54200 

11800000 

69.30 

78120 

2.80 

46453 

I287IOOO 

2.OI 

58.3 

87300 

9-45 

54800 

11500000 

•30 

22320 

14.60 

52127 

II436000 

2.01 

58-4 

86490 

8-73 

54200 

11900000 

•50 

7I-25 

36360 

10.30 

54867 

II482000 

2.OI 

58.4 

87120 

8.09 

54600 

11600000 

•33* 

71-75 

45360 

6.70 

50852 

12359000 

2.00 

59-8 

84870 

8.80 

54000 

11700000 

•So 

79-5° 

32760 

14.60 

49435 

IO7IOOO 

•27 

62 

23800 

4.71 

53920 

2720000 

.26 

63 

22950 

5.60 

54250 

223OOOO 

.29 

64 

24600 

5-24 

52840 

2510000 

•25 

60 

16640 

3-70 

52150 

2190000 

•25 

61 

2159° 

4.40 

5437° 

2510000 

•27 

66 

23750 

5-oo 

53890 

2840000 

•77 

71 

38970 

2.  IO 

35790 

I2OOOOO 

•75 

61.1 

56150 

7-1 

534oo 

1200000 

•75 

64.0 

55350 

8.7 

52600 

12OOOOO 

•72 

64-5 

45090 

5-4 

44900 

1800000 

•75 

61.0 

53360 

6.  i 

50700 

II500000 

-  /4 

63.0 

557*° 

1.49 

52900 

IT3OOOOO 

The  above  tables  show  the  results  of  tests  made  in  the  engi- 
.leering  laboratories  of  the  Massachusetts  Institute  of  Technol- 


TO  RSI  ON  A  L  S  TRENG  TH  OF  IV RO  UGHT-IRON  AND  S  TEEL.   545 

ogy  upon  the  torsional  strength  of  various  kinds  of  wrought-iron. 
The  figures  in  the  column  headed  "  Apparent  outside  fibre-stress  " 

Mr 

are  obtained    from  the  formula  /  =  -j-9  where  M  =  maximum 

twisting-moment,  r= outside  radius  of  shaft,  and  /  = polar  moment 
of  inertia  of  section.     Of  course  it  is  not  the  outside  fibre -stress. 


TORSIONAL  TESTS  OF  BESSEMER  STEEL. 


1 
O 

ri 

J8 

o 

c 

1 

+i  (-1 

S 

M 

|jti 

& 

*| 

0) 

a 

»—  1 

1 

jj 

f  +•>" 

d 
IjJ 

iF! 

I5 

f: 

S 

1 

bo 
J 

S 

CJ     • 

^1 

23  rt 

§* 

s 

1u 

11 

rt  o 

'S 

.3 

+3 

<u-£  c 

'O   W5 

Sf" 

«**"*  S 

•g|«S 

^  1  8 

ri 

!>« 

3 

J 

^S   ^ 

^  G 

CX^^g 

SM  ^ 

E  &  p 

S 

3 

S 

o 

1 

<w 

S 

I*" 

I    7C 

60  .00 

66960 

63632 

12418000 

ii  .88 

-1  •  /o 

en    7r 

30600 

"Ow«3'* 

7^031 

I  I243OOO 

I  -  "3Q 

oy  •  i  j 

fQ      CQ 

oy 
3762O 

/O^v)-1- 
712  CO 

A  i^  if  ^^WV-* 

12  CQ4OOO 

I  C  .OO 

2  .OO 

oy  •  ow 

56  .00 

o  / 
101520 

/     ow 

64630 

A^  ^yifwwvy 

1  1  82  OOOO 

7  ^5 

2  .OO 

55-Qo 

30 





100260 

63830 

10320000 

7.87 

2    O2 

c6  .  oo 

?lS 

I  I  1960 

*"o^S?r> 

I  IQQOOOO 

T     &C 

A  A     Ort 

2  4 

81240 

A  -  2  -/-\ 

I  O2  COOOO 

*      R^7 

jo    OI 

55  .00 

36 

112  COO 

'roT)  i  n 

I34IOOOO 

<;   76 

1   *  'w 

2  •  O2 

190  .  oo 

o 
144, 

oy 

I  34IOO 

82860 

I  l83OOOO 

D  *   / 

6    77 

96.00 

A  if  if 

75 

20160 

30400 

•'•O  ' 

56000 

84500 

12200000 

u  •  /  / 
16.30 

•5° 

94-00 

75 

19800 

29900 

5328o 

80400 

I220OOOO 

1-570 

•52 

93.00 

75 

21600 

31300 

53280 

77300 

IO70000O 

16.10 

•53 

94.00 

75 

21600 

30800 

52560 

7470° 

I09OOOOO 

15.80 

•  49 

60.00 

40 



43920 

66280 

IlSoOOOO 

8.60 

•5° 

58.00 

40 

14400 

21700 

44640 

67400 

H7OOOOO 

13.30 

•5° 

57  -5P 

40 

17000 

25600 

44820 

67600 

II900000 

ii  .50 

•5° 

57.60 

40 

17000 

25600 

45810 

69100 

II90OOOO 

9.90 

•5° 

59-oo 

40 

16000 

24100 

4545° 

68600 

Il6oOOOO 

10.  80 

•5° 

58.80 

40 

18000 

27200 

44460 

67100 

II700000 

10.50 

•5° 

59.10 

40 

l6j2OO 

24400 

45000 

67900 

II500000 

13.20 

•5° 

58.20 

40 

18000 

27200 

'  44920 

67800 

I  I  7OOOOO 

10.80 

•5° 

58.00 

40 

18000 

27200 

45540 

68700 

I  I  7OOOOO 

11.40 

546  APPLIED   MECHANICS, 

§  232.  Riveted  Joints.  —  The  most  common  way  of  uniting 
plates  of  wrought-iron  or  steel  is  by  means  of  rivets.  It  is, 
therefore,  a  matter  of  importance  to  know  the  strength  of  such 
joints,  and  also  the  proportions  which  will  render  their  efficie,n- 
cies  greatest;  i.e.,  that  will  bring  their  strength  as  near  as 
possible  to  the  strength  of  the  solid  plate. 

In  §  177  was  explained  the  mode  of  proportioning  riveted 
joints  usually  taught,  based  upon  the  principle  of  making  all 
the  resistances  to  giving  way  equal,  and  assuming,  as  the  modes 
of  giving  way,  those  there  enumerated.  This  theory  does  not, 
however,  represent  the  facts  of  the  case,  as  — 

i°.  The  stresses  which  resist  the  giving- way  are  of  a  more 
complex  nature  than  those  there  assumed,  so  that  the  efficiency 
of  a  joint  constructed  in  the  way  described  above  may  not  be 
as  great  as  that  of  one  differently  constructed ; 

2°.  The  effects  of  punching,  drilling,  and  riveting,  come  in 
to  modify  further  the  action  ;  and 

3°.  The  purposes  for  which  the  joint  is  to  be  used,  often  fix 
some  of  the  dimensions  within  narrow  limits  beforehand. 

In  order  to  know,  therefore,  the  efficiency  of  any  one  kind 
of  joint,  we  must  have  recourse  to  experiment.  And  here  again 
we  must  not  expect  to  draw  correct  conclusions  from  experi- 
ments made  upon  narrow  strips  of  plate  riveted  together  with 
one  or  two  rivets  ;  but  we  need  experiments  upon  joints  in  wide 
plates  containing  a  sufficiently  long  line  of  rivets  to  bring  into 
play  all  the  forces  that  we  have  in  the  actual  joint.  The  greater 
part  of  the  experiments  thus  far  made  have  been  made  upon 
narrow  strips,  with  but  few  rivets.  The  number  of  tests  of  the 
other  class  is  not  large,  and  of  those  that  have  been  made,  the 
greater  part  merely  furnish  us  information  as  to  the  behavior 
of  the  particular  form  of  joint  tested,  and  do  not  teach  us  how 
to  proportion  the  best  or  strongest  joint  in  any  given  plates,  as 
no  complete  and  systematic  series  of  tests  has  thus  far  been 
carried  out,  though  such  a  series  has  been  begun  on  the  govern- 
ment testing-machine  at  the  Watertown  Arsenal. 


RIVETED  JOINTS. 


The  only  tests  to  which  it  seems  to  the  writer  worth  while 
to  make  reference  here  are  : 

i°.  A  portion  of  those  made  by  a  committee  of  the  British 
Institution  of  Mechanical  Engineers,  inasmuch  as,  although  a 
very  large  part  were  made  upon  narrow  strips  with  but  few 
rivets,  nevertheless  a  portion  were  made  upon  wide  strips. 

2°.  The  tests  on  riveted  joints  that  have  been  made  on  the' 
government  testing-machine  at  Watertown  Arsenal. 

i°.  The  account  of  this  series  is  to  be  found  at  intervals 
from  1880  to  1885  inclusive,  with  one  supplementary  set  in  1888, 
in  the  proceedings  of  the  British  Institution  of  Mechanical 
Engineers;  but  as  all  except  the  supplementary  set  has  also 
been  published  in  London  Engineering,  these  latter  references 
will  be  given  here  as  follows : 

Engineering  for  1880,  vol.  29,  pages  no,  128,  148,  254,  300,  350. 

"    1881,  vol.  31,      "      427,  436,  458,  508,  588. 

"    1885,  vol.  39,      "      524. 

"    1885,  vol.  40,      "      19,43. 
Also,  Proc.  Brit.  Inst.  Mechl.  Engrs.,  Oct.  1888. 

2°.  The  second  series,  referred  to  above,  or  those  made  on 
the  government  testing-machine  at  Watertown  Arsenal,  are  to 
be  found  in  their  reports  of  the  following  years,  viz.,  1882, 
1883,  1885,  1886,  1887,  and  1895. 

3°.  Report  of  tests  of  structural  material  made  at  the 
Watertown  Arsenal,  Mass.,  June,  1891. 

While  it  is  from  tests  upon  long  joints  that  we  can  derive 
correct  and  reliable  information  to  use  in  practice,  and  hence 
while  the  experiments  already  made  give  us  a  considerable 
amount  of  information,  nevertheless  as  the  tests  have  not  yet 
been  carried  far  enough  to  furnish  all  the  information  we  need, 
and  to  settle  cases  that  we  are  liable  to  be  called  upon  to 
decide,  therefore,  before  quoting  the  above  experiments,  a  few 
of  the  rules  and  proportions  more  or  less  used  at  the  present 


54$  APPLIED   MECHANICS. 

time,  and  the  modes  of  determining  them,  will  be  first  ex- 
plained. 

In  this  regard  we  must  observe  that  practical  considerations 
render  it  necessary  to  make  the  proportions  different  when  the 
joint  is  in  the  shell  of  a  steam-boiler,  from  the  case  when  it  is 
in  a  girder  or  other  part  of  a  structure. 

In  the  case  of  boiler-work,  the  joint  must  be  steam-tight,  and 
hence  the  pitch  of  the  rivets  must  be  small  enough  to  render 
it  so :  whereas  in  girder-work  this  requirement  does  not  exist ; 
and  hence  the  pitch  can,  as  far  as  this  requirement  goes,  be 
made  greater. 

It  is  probable,  that,  with  good  workmanship,  we  might  be  able 
to  secure  a  steam-tight  joint  with  considerably  greater  pitches 
than  those  commonly  used  in  boiler-work ;  and  now  and  then 
some  boiler-maker  is  bold  enough  to  attempt  it. 

Some  years  ago  punching  was  the  most  common  practice  , 
but  now  drilling  has  displaced  punching  to  such  an  extent  that 
all  the  better  class  of  boiler-work  is  now  drilled,  and  drilling  is 
also  used  to  a  very  considerable  extent  in  girder-work.  When 
drilling  is  used,  the  plates,  etc.,  to  be  united  should  be  clamped 
together  and  the  holes  drilled  through  them  all  together.  In 
this  regard  it  should  be  said  : 

i°.  When  the  holes  are  drilled,  and  hence  no  injury  is  done 
to  the  metal  between  the  rivet-holes,  this  portion  of  the  plate 
comes  to  have  the  properties  of  a  grooved  specimen,  and  hence 
has  a  greater  tensile  strength  per  square  inch  than  a  straight 
specimen  of  the  same  plate,  as  the  metal  around  the  holes  has 
not  a  chance  to  stretch.  This  excess  tenacity  may  amount 
to  as  much  as  25  per  cent  in  some  cases,  though  it  is  usually 
nearer  10  or  12  per  cent,  depending  not  only  on  the  nature  of 
the  material,  but  also  on  the  proportions. 

2°.  When  the  holes  are  punched,  we  have,  again,  a  grooved 
specimen,  but  the  punching  injures  the  metal  around  the  hole, 
and  this  injury  is  greater  the  less  the  ductility  of  the  metal : 
thus,  much  less  injury  is  done  by  the  punch  to  soft-steel  plates 


RIVETED  JOINTS.  549 


than  to  wrought-iron  ones,  and  less  to  thin  than  to  thick  plates. 
This  injury  may  reach  as  much  as  35  per  cent,  or  it  may  be 
very  small.  Besides  this,  in  punching  there  is  liability  of  crack- 
ing the  plate,  and  of  not  having  the  holes  in  the  two  plates  that 
are  to  be  united  come  exactly  opposite  each  other.  A  number  of 
tests  on  the  tenacity  of  punched  and  drilled  plates  of  wrought- 
iron,  and  of  mild  steel,  made  on  the  government  testing-machine 
at  Watertown  Arsenal,  are  given  on  page  564  tt  seq. 

The  hardening  of  the  metal  by  punching  also  decreases  the 
ductility  of  the  piece. 

The  injury  done  by  punching  may  be  almost  entirely  re- 
moved in  either  of  the  following  ways  :  — 

i°.  By  annealing  the  plate. 

2°.  By  reaming  out  the  injured  portion  of  the  metal  around 
the  hole ;  i.e.,  by  punching  the  hole  a  little  smaller  than  is  de- 
sired, and  then  reaming  it  out  to  the  required  size. 

There  is  a  certain  friction  developed  by  the  contraction  of 
the  rivets  in  cooling,  tending  to  resist  the  giving  way  of  the 
joint ;  and  some  have  advocated  the  determination  of  the  safe 
load  upon  a  riveted  joint  on  the  basis  of  the  friction  developed, 
instead  of  on  the  basis  of  strength — notably  M.  Dupuy  in  the 
Annales  des  Fonts  et  Chausees  for  January,  1895  ;  but  this 
seems  to  the  author  an  erroneous  and  unsafe  method  of  pro- 
ceeding: i°,  because  tests  show  that  slipping  occurs  at  all 
loads,  beginning  at  loads  much  smaller  than  the  safe  loads  on 
the  joint ;  2°,  because  all  friction  disappears  before  the  break- 
ing load  is  reached. 

Hence  it  is  safer  to  disregard  friction  in  designing  a  tensile 
riveted  joint. 

The  shearing-strength  of  the  rivets  would  appear  to  be 
about  two  thirds  the  tensile  strength  of  the  rivet  metal. 

Before  proceeding  to  give  an  account  of  Kennedy's  tests, 
and  of  those  made  at  the  Watertown  Arsenal,  which  form  the 
principal  basis  for  determining  the  constants,  i.e.,  the  tearing- 
strength  of  the  plate,  the  shearing-strength  of  the  rivet  iron, 


550  APPLIED    MECHANICS. 

and  the  ultimate  compression  on  the  bearing  surface,  it  will  be 
best  to  outline  the  proper  method  of  designing  a  riveted  joint, 
and  for  this  purpose  a  discussion  of  a  few  cases  of  tensile  riveted 
joints,  as  given  by  Prof.  Peter  Schwamb,  will  be  given  by  way 
of  illustration. 

The  letters  used  will  be  as  follows,  viz. : 

d  =  diameter  of  driven 'rivet  in  inches ; 

t  =  thickness  of  plate  in  inches ; 

/!  —  thickness  of  one  cover-plate  in  inches ; 

fs  =  shearing-strength  of  rivet  per  square  inch ; 

ft  —  tearing-strength  of  plate  per  square  inch  ; 

fc  =  crushing-strength  of  rivet  or  plate  per  square  inch; 

/  —  pitch  of  rivets  in  inches ; 

pd  =  diagonal  pitch  in  inches ; 
/  =  lap  in  inches. 

In  every  case  of  a  tension-joint  we  begin  by  selecting  a 
repeating  section  and  noting  all  the  ways  in  which  it  may  fail. 
It  would  seem  natural,  then,  to  determine  the  diameter  of  the 
rivet  to  be  used  by  equating  the  resistance  to  shearing  and 
the  resistance  to  crushing,  and  in  some  cases  it  is  desirable  to 
adopt  the  resulting  diameter  of  rivet ;  but  there  are  also  many 
cases  where  there  is  good  reason  for  adopting  either  a  larger 
or  a  smaller  rivet,  and  others  where  there  is  good  reason  for 
determining  the  trial  diameter  in  some  other  way. 

Thus  we  may  find  that  the  rivet  which  presents  equal  re- 
sistance to  shearing  and  crushing  may  be  too  large  to  be  suc- 
cessfully worked,  or  it  may  require  a  pitch  too  large  for  the 
purposes  for  which  the  joint  is  to  be  used;  or,  on  the  other 
hand,  it  may  be  so  small  that  it  would  lead  to  a  pitch  too 
small  to  be  practicable ;  or  it  might,  in  a  complicated  joint, 
where  there  are  a  good  many  ways  of  possible  failing,  lead  to 
a  low  efficiency.  In  all  cases,  a  commercial  diameter  must  be 
selected. 


SINGLE-RIVETED    LAP-JOINT.  55 1 

Single-riveted  Lap-joint. — Repeating  section  containing 
one  rivet  may  fail  by — 

1°,  shearing  one  rivet.  Resistance  =^— t 

4 

2°,  tearing  the  plate.  Resistance  =ft (p—d)t. 

3°,  compression.  Resistance  —  fctd. 

Equating  i°  and  3°  gives     d  =  —  y  (i) 

71  ft 

A  larger  rivet  will  crush,  a  smaller  one  will  shear. 
The  diameter  given  by  (i)  will  frequently  be  found  to  be 
larger  than  can  be  successfully  worked. 

Equating  2°  and  3°  gives    p  —  d(\  +  y  ).  (2) 

Equating  i°  and  2°  gives    p  =  d  (i  +  ^  £}.  (3) 

4*  ft/ 

If  the  value  of  d  given  in  (i)  is  used,  then  (2)  and  (3)  give 
the  sameVesult.  If,  however,  a  different  value  of  d  is  used, 
then  the  pitch  should  be  determined  by  (2)  for  a  larger  and 
by  ($)  for  a  smaller  rivet. 

It  may  be  well  to  note  that  whenever  compression  fixes 
the  pitch,  the  computed  efficiency 

P~d_      f< 

P     -ft+f. 

is  independent  of  the  diameter  of  the  rivet,  and  that  this  is 
the  maximum  efficiency  obtainable  with  this  style  of  joint. 

SINGLE-RIVETED  DOUBLE-SHEAR  BUTT-JOINT. 

The  combined  thickness  of  the  two  cover-plates  should 
always  be  greater  than  /,  and,  this  being  the  case,  we  proceed 
as  follows :  • 


552  APPLIED    MECHANICS. 

Repeating  section  containing  one  rivet  may  fail  by — 
i  °,  shearing  one  rivet  in  two  places.  Resistance  =  /, . 

2°,  tearing  the  plate.  Resistance  =  ft(p  —  d)t. 

3°,  compression  Resistance  =fetd. 

Equating  i°  and  3°  gives      d=  —  j  (4) 

71  ft 

A  larger  rivet  will  crush,  a  smaller  one  will  shear. 

The  diameter  given  by  (4)  is  just  one  half  that  given  by  (i), 
f  nd  will  frequently  be  found  to  lead  to  a  pitch  too  small  to  use  in 
practice.  In  such  cases  we  should  use  a  larger  rivet. 

Equating  2°  and  3°  gives  p=d[i  +  j).  (5) 

v       It' 

Equating  i°  and  2°  gives  p=d(i+~j}.  (6) 

If  the  value  of  d  given  by  (4)  be  used,  then  (5)  and  (6)  give 
the  same  result.  If,  however,  a  different  value  of  d  be  used,  then 
the  pitch  should  be  determined  by  (5)  for  a  larger  and  by  (6)  for 
a  smaller  rivet. 

For  the  diagonal  pitch,  in  the  case  of  staggered  riveting,  we 
should  have,  at  least,  according  to  Kennedy's  sixth  conclusion 
(see  page  566)  2(pd—d)=^(p—d)  and  hence  pd  =  $p  +  $d. 

DOUBLE-RIVETED   LAP-JOINTS. 

Repeating  section  containing  two  rivets  may  fail  by — 

nd2 
i°,  shearing  two  rivet  sections.         Resistance  =  J8  — . 

2°,  tearing  plate  straight  across.      Resistance  =  }t(p—d)t. 
3°,  compression  on  two  rivets.         Resistance  =  jc(2td). 

Equating  i°  and  3°  gives  d=—  /.  (7) 

7T  /« 


EXAMPLE   OF  A    SPECIAL  JOINT.  553 

A  larger  rivet  will  crush,  a  smaller  one  will  shear. 

The  diameter  given  by  (7)  would  usually  be  found  too  large. 

Equating  2°  and  3°  gives  p=d(i+-j?J.  (8) 

Equating  i°  and  2°  gives  p=d  \i  H  --  ^/  .  (9) 

The  pitch  should  be  determined  by  (8)  for  a  larger  and  by  (9) 
for  a  smaller  rivet  than  that  given  by  (7). 

For  pd  we  should  have,  as  in  the  last  case,  according  to 
Kennedy,  pd 


EXAMPLE   OF   A  SPECIAL   JOINT. 

The  joint  shown  in  the  cut  is  one  where  a  part  of  the  rivets 
are  in  single  and  a  part  in  double  shear. 

Repeating    section    containing    five    rivet 
sections  may  fail  by— 

i°,  tearing  on  ab. 

Resistance  =  ft(p—  d)t. 

2°,  shearing  five  rivet  sections. 

, 
•Resistance  =  /a 


4 
3°,  tearing  on  ce,  and  shearing  one  rivet  on  ab. 

nd2 
Resistance  =  jt(p  —  zd)t  +  f8  —  . 

4 
4°,  tearing  on  ce,  and  crushing  one  rivet. 

Resistance  =  jt(p-  2d)  +  fad. 
5°,  crushing  two  rivets  and  shearing  one. 

Resistance  =fc(2td)  +  /,  —  . 
4 

6°,  crushing  on  three  rivets.     Resistance  =  fc(2td+tid). 
7°,  crushing  three  rivets,  where  /i  ^  /. 

Resistance  = 


554  APPLIED    MECHANICS. 

In  this  case,  we  should  so  proportion  the  joint  that  its  effi- 
ciency may  be  determined  from  its  resistance  to  tearing  along  ab. 
Hence  all  its  other  resistances  should  be  equal  to  or  greater  than 
this. 

Hence  equate  i°  and  3°,  and  calculate  the  resulting  diameter 
of  rivet,  which  will  generally  be  too  small,  and  hence  we  select 
a  larger  rivet,  so  that  3°  may  be  greater  than  i°. 

Having  fixed  the  diameter  of  rivet,  determine  the  pitch  in 
each  of  three  ways,  viz.,  by  equating  i°  and  2°,  by  equating  i° 
and  6°,  and  by  equating  i°  and  5°,  and  adopt  the  least  value  of  p. 

In  this  joint  as  used  fji d  >  f8 — ,  and  hence  6°  is  greater  than 


LAP, 

To  compute  the  lap,  the  following  method  is  a  good  one. 
Consider  the  plate  in  front  of  the  rivet  as  a  rectangular  beam 
fixed  at  the  ends  and  loaded  at  the  middle,  whose  span=d, 
breadth  =/  (for  cover-plate  t]),  depih  =  h=l—d/2.  Assume  for 
modulus  of  rupture  jt  and  for  center  load  W,  where 

i°.  When  rivet  fails  by  single  shear  W=js . 

4 

2°.  When  rivet  fails  by  double  shear  W=f8 — . 

3°.  When  rivet  fails  by  crushing   and  lap  in  plate  is  sought 


4°.  When  rivet  fails  by  crushing  and  lap  in  cover-plate  is 
sought  W  =  jctid. 


JOINTS  IN   THE    WEB   OF  A    PLA  TE   GIRDER. 


555 


JOINTS    IN    THE    WEB    OF    A   PLATE    GIRDER. 

While  no  experiments  on  the  strength  of  such  joints  have 
been  published,  the  constants  necessary  for  use  in  the  ordinary 
method  of  calculating  them  are  :  i°,  the  allowable  outside 
fibre-stress ;  2°,  the  allowable  shearing-stress  on  the  outer 
rivet ;  and,  3°,  the  allowable  compression  on  the  bearing- 
surface. 

As  an  example  of  the  usual  method  of  calculation  of  such 
a   joint,  let    us  consider   a  chain-riveted    butt-joint  with  two 
covering  strips  (as  shown  in  the  cut)  as  being  a  joint  in  the 
web  of  a  plate  girder  which  has  equal 
flanges,  and  let  us  determine  the  allow- 
able amount  of  bending-moment  which 
the  web  alone  (without  the  flanges)  can 
resist.       The   modifications    necessary 
when    the    flanges    are    unequal,    and 
hence  when  the  neutral  axis  is  not  at 
the  middle  of  the  depth,  will  readily 
suggest  themselves. 

The  stress  on  any  one  rivet  is  pro- 
portional   to    its   distance    from    the 

neutral  axis  of  the  girder,  and  hence,  in  this  case,  from  the 
middle  of  the  depth. 

Use  the  following  letters,  viz.: 

ft  —  allowable  stress  per  sq.  in.  at  outer  edge  of  web-plate  ; 
ft  =  allowable  shearing-stress  per  sq.  in.  on  outer  rivet ;  fc  = 
allowable  bearing-pressure  per  sq.  in.  on  outer  rivet ;  /  =  thick- 
ness of  plate  ;  h  —  total  depth  of  web-plate  ;  /^  =  total  depth 

of  girder  ;  d=  diameter  of  driven  rivet  ;  a  —  —    —  =    area     of 

4 

driven  rivet  section  ;  r  =  number  of  vertical  rows  on  each  side  ; 
2n  =  number  of  rivets  in  each  vertical  row  ;  yl  =  distance  from 


o     o 

0      0 

0      O 

O      0 

0      0 

0      0 

o    o 

0      0 

0      0 

0      0 

0      0 

o    o 

o     o 

0      0 

0      0 

0     0 

556  APPLIED    MECHANICS. 

neutral  axis  to  centre  of  nearest  rivet  ;  yt  =  distance  from 
neutral  axis  to  centre  of  second  rivet,  etc.,  etc.;jM  =  distance 
from  neutral  axis  to  centre  of  outer  rivet. 

Then,  for  allowable  bending-moment,  we  must  take  the 
least  of  the  three  following,  viz  : 

i°,  that  determined  from  the  shearing  fs  ; 

2°,  that  determined  from  the  compression  fc\ 

3°,  that  determined  from  maximum  fibre-stress  ft  ,  observ- 
ing that  if  f  =  greatest  allowable  fibre-stress  in  girder,  then 


To  determine  these  proceed  as  follows  : 


hence  allowable  stress  on  rivet  at  distance  ym  from  neutral  axis 


i/    •  — 


and  the  moment  of  this  stress  is 


Hence  greatest  allowable  moment  on  joint  for  shearing  is 

(,) 


JOINTS  IN   THE    WEB   OF  A    PLATE   GIRDER.  557 


2°.  Greatest  allowable  compression  on  outer  rivet  \sfctd\ 
hence  allowable  stress  on  rivet  at  distance ym  from  neutral  axis  is 


.  • 

y*      ' 


and  the  moment  of  this  stress  is 


Hence  greatest  allowable  moment  on  joint  for  compression  is 

-.  './i  •      *r   a          i       «r   *          i  i      *r    '•     t  \      / 

Jn 

3°.  The  section  of  the  plate  is  a  rectangle,  width  /  and 
height  //,  with  the  spaces  where  the  rivet-holes  are  cut  left  out. 
It  will  be  near  enough  to  take  for  the  stress  to  be  deducted  on 
account  of  the  rivet-hole  at  distance  ym  from  neutral  axis 


and  for  its  moment 


Hence  greatest  allowable  moment  on  joint  for  tearing  is 


558  APPLIED    MECHANICS. 


This  mode  of  calculation  for  (3)  would  seem  to  be  war- 
ranted from  the  fact  that  the  rivets  do  not  fill  the  holes, 
although  many  deduct  only  the  effect  of  the  holes  on  the  ten- 
sion side,  and  consider  that  those  on  the  compression  side  do  not 
weaken  the  metal.  The  greatest  allowable  bending-moment  on 
the  joint  is  the  smallest  of  (i),  (2),  and  (3),  and  it  is  plain  that,  in 
order  to  make  the  calculation,  we  need  to  know  what  to  use 

o  fs,  and/c,  or,  since  ft  =f-j-,  what  to  use  for  /,  fs,  and 


fe\  and  while  /"should  be  determined  from  the  tests  on  the 
transverse  strength  of  the  metal,  whether  wrought-iron  or  steel, 
the  best  evidence  we  have  as  to  the  proper  values  of  fs  and  fe 
is  furnished  by  the  tests  on  tension-joints,  which  have  already 
been  discussed. 

Moreover,  we  might  determine  the  diameter  of  rivet  by 
equating  (i)  and  (2),  but  we  should  generally  find  it  desirable 
to  use  a  larger  rivet,  and  then  we  should  determine  the  pitch 
by  equating  (2)  and  (3)  if  a  larger,  or  (i)  and  (2)  if  a  smaller, 
rivet  is  used. 

Moreover,  the  rivets  in  common  use  in  such  cases  are  either 
f"  or  I"  in  diameter. 


TESTS    OF    THE    COMMITTEE    OF    THE    BRITISH    INSTITUTION    OF 
MECHANICAL    ENGINEERS. 

The  Committee  on  Riveted  Joints  of  the  British  Institu- 
tion of  Mechanical  Engineers  consisted  of  Messrs.  W.  Boyd, 
W.  O.  Hall,  A.  B.  W.  Kennedy,  R.  N.  J.  Knight,  W.  Parker, 
R.  H.  Twedell,  and  W.  C.  Unwin. 


RI VE  TED  JOIN  TS.  559 


Before  beginning  operations  Prof.  Unwin  was  asked  to 
prepare  a  preliminary  report,  giving  a  summary  of  what  had 
already  been  done  by  way  of  experiment,  and  also  to  make 
recommendations  as  to  the  course  to  be  pursued  in  the  tests. 

This  preliminary  report  is  contained  in  vol.  xxix.  of  Engineer- 
ing, on  the  pages  already  cited.  In  regard  to  its  recommenda- 
tions it  is  unnecessary  to  speak  here,  as  the  records  of  the  tests 
show  what  was  done ;  but  in  regard  to  the  summary  of  what 
had  been  done,  it  may  be  well  to  say  that  he  gives  a  list  of 
forty  references  to  tests  that  had  been  made  before  1880,  be- 
ginning with  those  of  Fairbairn  in  1850,  and  ending  with  some 
made  by  Greig  and  Eyth  in  1879,  together  with  a  brief  account 
of  a  number  of  them. 

Almost  all  of  this  work  was  done,  however,  with  small  strips 
with  but  few  rivets,  and  will  not  be  mentioned  here.  Inas- 
much, however,  as  Fairbairn's  proportional  numbers  have  been 
very  extensively  published,  and  are  constantly  referred  to  by 
the  books  and  by  engineers,  it  may  be  well  to  quote  a  portion 
of  what  Unwin  says  in  that  regard,  as  follows  : 

"  The  earliest  published  experiments  on  riveted  joints,  and 
probably  the  first  experiments  on  the  strength  of  riveting  ever 
made,  are  contained  in  the  memoir  by  Sir  Wm.  Fairbairn  in  the 
Transactions  of  the  Royal  Society. 

"  The  author  first  determined  the  tenacity  of  the  iron,  and 
found,  for  the  kinds  of  iron  experimented  upon,  a  mean  tenacity 
of  22.5  tons  per  square  inch  with  the  stress  applied  in  the 
direction  of  the  fibre,  and  23  with  the  stress  across  it.  That 
the  plates  were  found  stronger  in  a  direction  at  right  angles  to 
that  in  which  they  were  rolled  is  probably  due  to  some  error 
in  marking  the  plates. 

"  Making  certain  empirical  allowances,  Sir  Wm.  Fairbairn 
adopted  the  following  ratios  as  expressing  the  relative  strength 
of  riveted  joints  : 


560  APPLIED  MECHANICS. 

Solid  plate 100 

Double-riveted  joint 70 

Single-riveted  joint 50 

These  well-known  ratios  are  quoted  in  most  treatises  on  rivet- 
ing,  and  are  still  sometimes  referred  .to  as  having  a  considerable 
authority. 

"  It  is  singular,  however,  that  Sir  Wm.  Fairbairn  does  not 
appear  to  have  been  aware  that  the  proportion  of  metal 
punched  out  in  the  line  of  fracture  ought  to  be  different  in 
properly  designed  double  and  single  riveted  joints.  These 
celebrated  ratios  would  therefore  appear  to  rest  on  a  very 
unsatisfactory  analysis  of  the  experiments  on  which  they  are 
based.  Sir  Wm.  Fairbairn  also  gives  a  well-known  table  of 
standard  dimensions  for  riveted  joints.  It  is  not  very  clear 
how  this  table  has  been  computed,  and  it  gives  proportions 
which  make  the  ratio  of  tearing  to  shearing  area  different  for 
different  thicknesses  of  plate.  There  is  no  good  reason  for 
this." 

As  to  the  tests  which  constitute  the  experimental  work  of 
the  committee,  these  were  made  by  or  under  the  direction  of 
Pi*of.  A.  B.  W.  Kennedy,  of  London.  Steel  plates  and  steel 
rivets  were  used  throughout,  the  steel  containing  about  0.18  per 
cent  of  carbon,  and  having  a  tensile  strength  varying  from 
about  62000  to  about  70000  pounds  per  square  inch,  and  hence 
being  a  little  harder  than  would  correspond  to  our  American 
ideas  of  what  is  suitable  for  use  in  steam-boilers.  The  greater 
portion  of  the  work  was  performed  by  the  use  of  a  testing- 
machine  of  looooo  pounds  capacity,  and  hence  one  which  did 
not  admit  of  testing  wide  strips  with  a  sufficient  number  of 
rivets  to  correspond  to  the  cases  which  occur  in  practice; 
indeed,  only  eighteen  of  the  tests  were  made  on  such  strips. 
Nevertheless,  a  brief  summary  of  what  was  done  will  be  given 
here,  though  some  of  the  conclusions  which  he  drew  are  aL 


RIVETED  JOINTS.  561 


ready,  and  others  are  liable  to  be,  proved  untrue  by  tests  of 
wide  strips.  The  tests  made  by  Prof.  Kennedy  up  to  1885 
consisted  of  fourteen  series  numbered  I  to  V,  VA  and  VI  to 
XIII,  and  covering  290  experiments,  64  on  punched  or  drilled 
plates,  97  on  joints,  44  on  the  tenacity  of  the  plates  used  in 
the  joints,  33  on  the  tenacity  and  shearing-resistance  of  the 
rivet-steel  used  in  the  joints,  and  the  remaining  52  on  various 
other  matters. 

The  first  three  series  were  upon  the  tenacity  of  the  steel 
used,  and  showed  it  to  be,  as  stated,  from  62000  to  70000  pounds 
per  square  inch,  with  an  ultimate  elongation  of  23  to  25  per 
cent  in  a  gauged  length  of  ten  inches ;  the  tenacity  of  the 
rivet-steel  being  practically  the  same  as  that  of  the  plates. 
The  fourth  series  showed  the  shearing-strength  of  the  rivet- 
steel  to  be  about  55000  pounds  per  square  inch  when  tested  in 
one  way,  and  59000  pounds  per  square  inch  when  tested  in 
another  way  which  corresponded,  as  Kennedy  claims,  better 
to  the  conditions  of  a  rivet,  though  neither  was  by  using  a 
riveted  joint. 

The  tests  of  series  V  and  VA  were  made  upon  pieces  of 
plate  which  had  been  punched  or  drilled,  in  other  words,  on 
grooved  specimens  ;  and,  as  might  be  expected,  these  specimens 
showed  invariably  an  increase  in  tensile  strength  over  the 
straight  specimens.  In  the  J"  and  fV'  plates  drilled  with  holes 
i  inch  in  diameter  and  2  inches  pitch,  the  net  metal  between 
the  holes  had  a  tenacity  11  to  12  per  cent  greater  than  that  of 
the  untouched  plate.  Even  with  punched  holes  the  metal  had 
a  similar  excess  of  tenacity  of  over  6  per  cent.  The  remaining 
eight  series,  VI  to  XIII  inclusive,  were  made  on  riveted  joints, 
the  first  five  on  single-riveted  lap-joints,  and  the  last  three, 
or  XI,  XII,  and  XIII,  on  double-riveted  lap  and  butt  joints. 

Series  VI  was  made  on  twelve  joints  in  f-inch  plates  which 
contained  only  two  rivets  each,  the  proportions  not  being  in- 
tended to  be  those  of  practice,  but  such  as  should  give,  to 


562  APPLIED  MECHANICS. 

some  extent,  limiting  values  for  the  resistances  of  the  plate  to 
tearing,  and  of  the  rivets  to  shearing  and  pressure.  The  results 
were  rather  irregular;  and  the  main  conclusion  which  he  drew, 
was,  that  if  the  joint  is  not  to  break  by  shearing,  the  ratio  of 
the  tearing  to  the  shearing  area  must  be  computed  on  a  much 
lower  value  of  shearing-strength  per  square  inch  than  the  ex- 
periments of  series  IV  had  shown ;  indeed,  some  of  the  joints 
of  series  VI  gave  way  by  shearing  the  rivets  at  loads  no  greater 
than  36000  pounds  per  square  inch  of  shearing-area. 

Series  VII  was  made  upon  six  (single-riveted  lap)  joints  in 
f-inch  plate,  with  only  three  f-inch  rivets  in  each  joint,  and 
with  varying  pitch  and  lap ;  all  these  joints  breaking  by  shear- 
ing the  rivets.  His  conclusion  from  these  tests  was,  that  the 
lap  need  not  be  more  than  1.5  times  the  diameter  of  the  rivet. 

Series  VIII  was  made  on  eighteen  (single-riveted  lap)  joints 
in  six  sets  of  three  each,  and  these  are  the  only  single-riveted 
lap-joints  which  he  tested,  having  as  many  as  seven  rivets  each. 
The  results  are  given  in  the  accompanying  table. 

Before  giving  the  table,  it  may  be  said  that  No.  652  was  in- 
tended to  have  such  proportions  as  to  be  equally  likely  to  give 
way  by  tearing  or  by  shearing,  the  intensity  of  the  shearing- 
strength  being  assumed  as  two-thirds  that  of  the  tensile 
strength  of  the  steel,  while  the  bearing-pressure  per  square 
inch  was  intended  to  be  about  7.5  per  cent  greater  than  the 
tension.  No.  653  was  proportioned  with  excess  of  shearing  or 
rivet-area,  No.  654  with  defect  of  shearing-area,  No.  655  with 
excess  of  tearing  or  plate  area,  No.  656  with  defect  of  tearing- 
area,  and  No.  657  with  excess  of  bearing-pressure,  the  different 
proportions  being  arrived  at  by  varying  the  pitch  and  diameter 
of  the  rivets,  and,  in  the  case  of  657,  the  thickness  of  the  plate 
also.  The  margin  (or  lap  minus  radius  of  rivet)  was  f  inch  in 
each  case.  The  following  table  will  show  how  far  these  inten- 
tions were  realized,  and  further  comments  will  be  deferred  till 
later. 


RIVETED  JOINTS. 


563 


P!l°S    J°   5U33  a3d  5u!of 
jo    q^Susj^s      iBuopaodoaj 

M 

in 

00 

in 

in 

0 
oo" 
in 

CO 

CO 

] 

Shearing. 

rt  rt 

O  be  bi> 

ice 
H'-'~ 

Shearing. 

Shearing. 

»L  ^ 

c  c 

ill 

Shearing. 

Shearing. 

•pajjno 
pBoq-Sui3iB3jg  jo  uopaodoaj 

.j 

U      CO 

o.  ^ 

OJ 
in 

J 

I 

0 

N 

q 

O 

*j    r*' 
3    N 

•qoiq  3JBnbg  -lad  spunoj 
juiof  uaqM.  djnssdj  j  SuiJBsg 

1 

S 

° 

o 

tn 

1 

o 

rl- 
eo 

o 

in 

0? 
00 

^  8 
«  g 

s 

'B3JV  SuiJBSg 

CT       • 

| 

N 

2 

CO 

M 

0 
in 

I 

1  ^ 

in 
U3 
rt 
.j 

•sjjojq  luiof  usqAv  B3jy 
SuiJBsqs  jo  'uj  'bg  Jad  ssaa^s 

if 

I 

o 

in 

T 

S, 

€ 

1 

5     0 
X    co 

€ 

tNERAl 

•B3jy  3uiJB3qs 

.a  £• 

! 

O 

CO 

O 

CO 

M 
CO 
CO 

CO 

rt- 

CO 

2 

HH 

•asjoaq  ?u;of  usqM.  BSJV 
SUUBSXJO  'ui  'bs  Jad  ss3j;s 

4 

1 

1 

vO 

O 

1 
oo 

1 

01 

O 
01 

> 

W 

« 

-"•*->• 

O     *"** 

CO 

oi 

R 

in 

01 

M 

CO 
04 

H 

V} 

•B3JV 

3uuB3X  01  SujjBsg  jo  op-e^[ 

0 

M 

CO 

o* 

6 

CO 
CO 

o 

O 

CO 

d 

'B3jy  S\Jl 
-JB3X    Ol  3uiJB3qS  JO  OpB^ 

^ 

tn 

vO 
CO 

? 

rl 

I~N. 

«-« 

jl 

0 

CO 

0* 

0* 

O 
00 
CO 

0 

1 

0 

d 

"S3I°H  J°  ^35Jd 

•S'       M- 

00 

M 

^ 

rj- 

5 

'S31°H  P3II!JQ  J°  JajatQBiQ 

—   d 

vO 
CO 

0 

in 

6 

d 

00 

d 

d 

.^awoi 

c    «? 

M 

g 

8 

oi 

i-i 

tn 

d 

oJ 

•snarapads  jo  -oN 

CO 

CO 

co 

CO 

CO 

CO 

•JS3X  JO  '0>I 

tn 

CO 

m 

0 

tn 

0 

tn 

0 

in 

tn 

tn 

O 

564  APPLIED   MECHANICS. 

Series  IX  was  made  on  twenty-one  joints  in  f-inch  plate 
(each  containing  only  two  rivets)  designed  in  a  manner  similar 
to  series  VIII,  while  three  were  afterwards  made  from  some 
of  the  broken  plates,  with  as  heavy  rivets  as  it  was  deemed 
possible  to  make  tight. 

From  these  tests  Kennedy  thinks  it  fair  to  conclude  — 

I °.  That  the  efficiency  of  a  single-riveted  lap-joint  in  a  |-inch 
plate  cannot  be  greater  than  50  per  cent,  unless  rivets  larger 
than  i.i  inch  are  used  ;  and  he  also  calls  attention  to  the  fact 
that,  as  he  claims,  strength  is  gained  by  putting  more  metal  in 
the  heads  and  ends  of  the  rivets,  claiming  that  it  will  make 
also  a  tighter  joint  for  boiler-work. 

Series  X  was  made  on  eight  single-riveted  lap-joints  in 
J-inch  and  f-inch  plate,  made  from  the  broken  specimens  of 
series  V  and  VA ;  they  also  had  only  two  rivets  each.  These 
joints  were  made  with  a  view  of  investigating  the  effect  of 
more  or  less  bearing-pressure.  He  claims  that  high  bearing- 
pressure  induces  a  low  shearing-strength  in  the  rivets,  and  that 
the  bearing-pressure  should  not  exceed  about  96000  pounds  per 
square  inch ;  also,  that  when  a  large  bearing-pressure  is  used, 
the  "  margin  "  should  be  extra  large  to  prevent  distortion,  and 
consequent  local  inequalities  of  stress  ;  also,  that  smaller  bearing- 
pressures  do  not  much  affect  the  strength  of  the  joint  one  way 
or  the  other. 

Series  XI  was  made  upon  twelve  specimens  of  double-riveted 
joints ;  three  being  lap-joints  in  f-inch  plate,  three  lap-joints  in 
f-inch  plate,  three  butt-joints  with  two  equal  covers  in  f-inch 
plate,  and  three  butt-joints  with  two  equal  covers  in  f-inch 
plate.  Kennedy  designed  these  joints  with  a  view  to  their 
being  equally  likely  to  fail  by  tearing  or  by  shearing.  His  as- 
sumptions and  the  results  of  the  tests  are  all  given  in  the  fol- 
lowing table : 


RIVETED  JOINTS. 


565 


SERIES  XI.     DOUBLE-RIVETED  LAP  AND  BUTT  JOINTS— AVERAGES. 


!| 

•5  -Q 

II 

| 

1! 

Mi 

U     *-• 

J*J§ 

8  « 

|l 

&  § 

S  £ 
&  a 

1 

Ss 

Iv, 

CO    0     V 

PH    rt 

c 

0 

W 

4^ 

3 

be   3 

O    V 

C/)          ^ 

_C 

»™% 

J2 

> 

"^   C/T 

.S# 

O 

he  2 

u   'O 

o  J5 

bo  g 

U 

E 

"o 

"S 

h 

jU 

£ 

o 

£ 

Q  > 
c 

II 

•s  -G    v 

SJ3 

W          CL 

G    o     U 
*Q    53     cti 

Is 

o 

g 

-0    S 

T3     *• 

S2 

^ 

£  £ 

H   «*  £ 

C/2    u     rt 

fr 

V 

c 
J4 

fl' 

W     > 

a  5 

> 

g 

I  s 

•s-S  S 

•s  ^  ° 

2   3   a 

--;    cfl     rt 
rt    3     w 

"3  ™  jj 

e 
u 

rt 

w     O 

2 

«P 

0     > 
X    O 

S  ^  5 

3   cr  o 

o  C/3  '3 

H  jr  ^ 

§  o.  2 

H 

Q 

< 

•< 

H 

Q 

PQ 

<5 

<; 

•< 

H 

In. 

In. 

Lbs. 

Lbs. 

In. 

In. 

£ 

Lbs. 

Lbs. 

Lbs. 

Lbs. 

% 

LAP-JOINTS. 

f 

0.8 

70560 

51970 

2.9 

2.15 

29 

89609 

75150 

53920 

91530 

80.8 

f 

i.i 

70560 

51970 

3-1 

2-45 

35 

Low 

69910 

49710 

58910 

70.8 

BUTT-JOINTS  WITH  TWO  COVERS. 

1 

0.7 

70560 

34720 

2-75 

2.00 

27 

68000 

33780 

94710 

80.2 

1 

i.i 

67200 

42560 

4.4 

3-18 

26 

100800    59290 

37650 

88460 

71-3 

Series  XII  contains  the  same  joints  as  series  XI,  the  strained 
ends  having  been  cut  off,  and  the  rest  redrilled  and  riveted  by 
means  of  Mr.  Twedell's  hydraulic  riveter;  and  series  XIII  con- 
tained  the  same  joints  treated  a  second  time  in  the  same  way. 
These  experiments,  so  far  as  they  went,  showed  no  gain  in 
ultimate  strength  to  result  from  hydraulic  as  compared  with 
hand-riveting ;  but  it  was  found  that,  through  a  misunderstand- 
ing, they  had  been  riveted  up  at  a  pressure  much  lower  than 
that  intended  by  Mr.  Twedell. 

On  the  other  hand,  the  load  at  which  visible  slips  occurred 
was  about  twice  as  much  greater  with  hydraulic  as  with  hand 
riveting. 


$66  APPLIED  MECHANICS. 


KENNEDY  S   CONCLUSIONS. 

The  following  are  a  portion  of  what  he  gives  as  his  con- 
elusions : 

i°.  The  metal  between  the  rivet-holes  had  a  considerably 
greater  tensile  resistance  per  square  inch  than  the  unperfo- 
rated  metal. 

2°.  In  single-riveted  joints,  with  the  metal  that  he  used,  he 
assumed  about  22  tons  (49280  Ibs.)  per  square  inch  as  the  shear- 
ing-strength  of  the  rivet-steel  when  the  bearing-pressure  is 
below  40  tons  (89600  Ibs.)  per  square  inch.  In  double-riveted 
joints  with  rivets  of  about  f-inch  diameter  we  can  generally 
assume  24  tons  (53760  Ibs.)  per  square  inch,  though  some  fell 
to  22  tons  (49280  Ibs.). 

3°.  He  advises  large  rivet  heads  and  ends. 

4°.  For  ordinary  joints  the  bearing-pressure  should  not  ex- 
ceed 42  or  43  tons  (94000  or  96000  Ibs.)  per  square  inch.  For 
double-riveted  butt-joints  a  higher  bearing-pressure  may  be 
allowed ;  the  effect  of  a  high  bearing-pressure  is  to  lower  the 
shearing-strength  of  the  steel  rivets. 

5°.  He  advises  for  margin  the  diameter  of  the  hole,  except 
in  double-riveted  butt-joints,  where  it  should  be  somewhat 
larger. 

6°.  In  a  double-riveted  butt-joint  the  net  metal,  measured 
zigzag,  should  be  from  30  to  35  per  cent  greater  than  that  meas- 
ured straight  across,  i.e.,  the  diagonal  pitch  should  be  —  p  -\ — > 

o          j 
where/  —  transverse  pitch  and  d-=-  diameter  of  rivet-hole. 

7°.  Visible  slip  occurs  at  a  point  far  below  the  breaking- 
load,  and  in  no  way  proportional  to  that  load. 

Kennedy  thinks  that  these  tests  enable  him  to  deduce  rules 
for  proportioning  riveted  joints,  and  the  following  are  his  rules, 
viz. : 


RIVETED  JOINTS.  567 


(a)  For  single-riveted  lap-joints  the  diameter  of  the  hole 
should  be  2\  times  the  thickness  of  the  plate,  and  the  pitch  of 
the  rivets  2f  times  the  diameter  of  the  hole,  the  plate-area  being 
thus  71  per  cent  of  the  rivet-area.  If  smaller  rivets  are  used, 
as  is  generally  the  case,  he  recommends  the  use  of  the  follow- 
ing  formula  : 


where  /  =  thickness  of  plate,  d—  diameter  of  rivet,  and/  = 

pitch. 

For  30-ton  (67200  Ibs.)  plate,  and  22-ton  (49280  Ibs.)  rivets,  a  =  0.524 
For  28-ton  (62720  Ibs.)  plate,  and  22-ton  (49280  Ibs.)  rivets,  a  =  0.558 
For  30-ton  (67200  Ibs.)  plate,  and  24-ton  (53760  Ibs.)  rivets,  a  =  0.570 
For  28-ton  (62720  Ibs.)  plate,  and  24  ton  (53760  Ibs.)  rivets,  a  =  0.606 

Or,  as  a  mean,  a  =  0.56. 

(&)  For  double-riveted  lap-joints  he  claims  that  it  would  be 
desirable  to  have  the  diameter  of  the  rivet  2\  times  the  thick- 
ness of  the  plate,  and  that  the  ratio  of  pitch  to  diameter  of 
hole  should  be  3.64  for  3<D-ton  (67200  Ibs.)  plate  and  22-ton 
(49280  Ibs.)  or  24-ton  (53760  Ibs.)  rivets,  and  3.82  for  28-ton 
(62720  Ibs.)  plate. 

Here,  however,  it  is  specially  likely  that  this  size  of  rivet 
may  be  inconveniently  large,  and  then  he  says  they  should  be 
made  as  large  as  possible,  and  the  pitch  should  be  determined 
from  the  formula  to 

/  =  "y  +  4 
where, 

For  30-ton  (67200  Ibs.)  plate,  and  24-ton  (53760  Ibs.)  rivets,  «=  x.if 
For  28-ton  (62720  Ibs.)  plate,  and  22-ton  (49280  Ibs.)  rivets,  a  =  1.16 
For  3o-ton  (67200  Ibs.)  plate,  and  22-ton  (49280  Ibs.)  rivets,  a  =  1.06 
For  28-ton  (62720  Ibs.)  plate,  and  24-ton  (53760  Ibs.)  rivets,  a  =  1.24 


568  APPLIED  MECHANICS. 

(c)  For  double-riveted  butt-joints  he  recommends  that  the 
diameter  of  the  hole  should  be  about  1.8  times  the  thickness 
of  the  plate,  and  the  pitch  4.1  times  the  diameter  of  the  hole, 
and  that  this  latter  ratio  be  maintained  even  when  the  former 
cannot  be. 

Two  of  the  principal  participants  in  the  discussion  of  the 
report  were  Mr.  R.  Charles  Longridge  and  Prof.  W.  C.  Unwin. 

Mr.  Longridge  was  of  the  opinion  that  wider  strips  with 
more  rivets  should  have  been  used  ;  that  holding  the  specimens 
in  the  machine  by  means  of  a  central  pin  at  each  end  was  not 
the  best  method ;  that  the  results  obtained  from  specimens 
which  had  been  made  from  the  remnants  of  other  fractured 
specimens  were  at  least  questionable,  for,  even  if  the  plate  had 
not  been  injured,  the  ratio  of  the  length  to  the  width  of  the 
narrowest  part  was  different  after  the  strained  ends  were  cut 
off  from  what  it  was  before  ;  that  machine-riveting  should  have 
been  adopted  throughout  instead  of  hand-riveting,  as  it  is  not 
possible  to  secure  uniformity  with  the  latter  even  were  it  all 
done  by  the  same  man,  as  he  would  be  more  tired  at  one  time 
than  at  another ;  that  experiments  should  be  made  to  determine 
the  effect  of  different  sizes  and  different  shapes  of  heads,  as 
well  as  of  different  pressures  upon  the  load  causing  visible  slip  , 
and  that  experiments  should  be  made  upon  chain-riveting,  as 
he  thought  the  chain-riveted  joint  would  show  a  greater  effi- 
ciency than  the  staggered. 

Professor  Unwin  said  : 

i°.  In  examining  the  results  to  ascertain  how  far  a  variation 
from  the  best  proportions  was  likely  to  affect  the  strength  of 
the  joint,  he  found  that  while  the  ratio  of  rivet  diameter  to 
thickness  of  plate  varied  21  per  cent,  the  ratio  of  shearing  to 
tearing  area  30  per  cent,  and  the  ratio  of  crushing  to  tearing 
area  34  per  cent,  the  efficiency  of  the  weakest  joint  was  only 
six  per  cent  less  than  that  of  the  strongest,  or,  in  other  words, 


RIVETED  JOINTS.  569 


the  whole  variation  of  strength  was  only  1 1  per  cent  of  the 
strength  of  the  weakest  joint. 

2°.  With  reference  to  the  effect  which  the  crushing-pres- 
sure on  the  rivet  produced  upon  the  strength  of  the  joint, 
there  were  some  old  experiments,  which  showed  that',  when 
the  bearing-pressure  on  the  rivet  became  very  large  there  was 
a  great  diminution  in  the  apparent  tenacity  of  the  plate  in 
the  case  of  riveted  joints  in  iron.  Why  should  the  crushing- 
pressure  affect  either  the  tenacity  of  the  plate  or  the  shearing 
resistance  of  the  rivet?  He  believed  that  it  did  not  really 
affect  either.  What  happened  was  that,  if  the  crushing-pres- 
sure exceeded  a  certain  limit,  there  was  a  flow  of  the  metal, 
and  the  section  which  was  resisting  the  load  was  diminished. 
Either  the  section  of  the  plate  in  front  of  the  rivet,  if  the  plate 
was  soft,  or  the  section  of  the  rivet  itself,  if  the  rivet  was  soft, 
became  reduced. 

3°.  He  thought  that  the  point  at  which  visible  slip  began 
was  the  initial  point  at  which  the  friction  of  the  plates  was 
overcome,  and  of  course  was  greater  the  greater  the  grip 
upon  the  plates,  and  hence  greater  in  machine  than  in  hand 
riveting.  In  some  cases  with  hydraulic  riveting  loads  were  got 
as  high  as  10  tons  (22400  Ibs.)  per  square  inch  of  rivet  section 
before  slipping  began. 

4°.  In  regard  to  the  rules  for  proportioning  riveted  joints, 
he  preferred  to  distinguish  the  joints  as  single-shear  and  double- 
shear  joints,  and  then  we  have  the  following  three  equations : 
one  by  equating  the  load  to  the  tearing-resistance  of  the  plates, 
a  second  by  equating  it  to  the  shearing-resistance  of  the  rivets, 
and  a  third  by  equating  it  to  the  crushing  resistance ;  these 
three  determining  the  thickness  of  the]  plate,  the  diameter  of 
the  rivet,  and  the  pitch. 

By  taking  the  crushing  as  double  the  tenacity,  we  should 
obtain  for  single  shear  d  =  2.57*,  and  for  double-shear,  d  = 


57°  APPLIED  MECHANICS. 

In  a  single-shear  joint  the  rivet  cannot  generally  be  made 
so  big,  and  in  the  double-shear  it  could  not  always  be  made  so 
small,  hence  the  rivet  diameter  is  chosen  arbitrarily,  and  then 
the  single-shear  joint  is  proportioned  by  the  equations  for  shear- 
ing  and  tearing,  no  attention  being  paid  to  the  crushing,  while 
the  double-shear  joint  is  proportioned  by  the  equations  for 
crushing  and  tearing,  no  attention  being  paid  to  the  shearing. 

5°.  The  general  drift  of  the  report  was  to  advocate  the  use 
of  larger  rivets.  Whether  this  could  be  done  or  not,  he  could 
not  say.  For  lap-joints  it  would  increase  the  strength,  whereas 
for  double-shear  joints  he  was  not  sure  that  it  would  not  be 
better  to  diminish  the  size  of  the  rivet,  and  hence  the  crushing, 
pressure. 

This  report  has  been  given  so  fully  because  it  emanates 
from  a  committee  of  the  British  Institution  of  Mechanical 
Engineers;  but  inasmuch  as  series  VIII  is  the  only  one  where 
wide  strips  were  used,  it  seems  to  the  writer  that  any  conclu- 
sions which  may  be  drawn  from  any  of  the  other  tests  given 
in  the  report  require  confirmation  by  tests  on  wide  strips  with 
more  rivets,  before  being  accepted  as  true. 

Government  Experiments. — The  references  to  these  experi- 
ments have  been  mentioned  on  page  ooo. 

Those  included  in  the  first  five  of  the  volumes  mentioned 
may  be  divided  into  three  parts:— 

i°.  Those  contained  in  the  first  two  Executive  Documents 
mentioned  above. 

2°.  Those  contained  in  the  third  and  fourth. 


RIVETED  JOINTS. 


3°.  Those  contained  in  the  fifth. 

Summaries  of  these  sets  of  tests  will  be  given  here  in  their 
order,  as  each  set  was  made  with  certain  special  objects  in 
view,  and,  if  not  all,  at  any  rate  the  i°  and  2°,  form,  as  has  been  al- 
ready stated,  the  first  portion  of  a  systematic  series ;  and  it  seems 
to  the  author  that,  although  the  series  are  not  yet  completed, 
yet  these  tests  themselves  furnish  more  reliable  information  in 
regard  to  the  behavior  and  the  strength  of  joints  than  any  other 
experiments  that  have  been  made,  and  that  the  figures  them- 
selves furnish  the  engineer  with  the  means  of  using  his  judg- 
ment in  many  cases  where  he  had  no  reliable  data  before. 

A  perusal  of  the  tables  will  give  a  good  idea  of  the  shear- 
ing-strength per  square  inch  of  the  rivet  iron,  which  is  seen  to 
be  less  than  the  tensile  strength  of  the  solid  plate  ;  also  the 
effect  on  strength  of  the  plates  due  to  the  entire  process  of 
riveting,  punching,  drilling,  and  driving  the  rivets ;  also  the 
efficiencies  of  the  joints  tested. 

One  of  the  strongest  single-riveted  joints  tested  was  a  single- 
riveted  lap-joint  with  a  single  covering-strip. 

The  apparent  anomaly  of  the  punched  plates  in  a  few  cases, 
showing  a  greater  strength  than  the  drilled  plates,  is  explained 
by  Mr.  Howard  to  be  due  to  the  strengthening  effect  of  cold- 
punching  combined  with  smallness  of  pitch,  inasmuch  as  then 
the  masses  of  hardened  metal  on  the  two  sides  re-enforce  each 
other. 

Further  than  this,  the  student  is  left  to  study  the  figures 
themselves  as  to  the  effect  of  different  proportions,  etc. 

In  regard  to  the  first  series,  i.e.,  those  contained  in  the  first 
two  Executive  Documents  mentioned,  it  is  stated  in  the  report 
that  — 

i°.  "  The  wrought-iron.  plate  was  furnished  by  one  maker 
out  of  one  quality  of  stock." 

2°  "  The  steel  plates  were  supplied  from  one  heat,  cast  in 
ingots  of  the  same  size;  the  thin  plates  differing  from  the 


5/2  APPLIED  MECHANICS. 

thicker  plates  only  in  the  amount  of  reduction  given  by  the 
rolls." 

The  modulus  of  elasticity  of  the  metal  was,  iron  plate^ 
31970000  Ibs. ;  steel  plate,  28570000  Ibs. 

In  the  tabulated  results,  the  manner  of  fracture  is  shown 
by  sketches  of  the  joints,  and  is  further  indicated  by  heavy 
figures  in  columns  headed  "  Maximum  Strains  on  Joints,  in  Jbs., 
per  Square  Inch." 


RIVETED  JOINTS. 


573 


•juiof  jo  Xouaptjjg 


°Q 

^O 


vr> 


9 

VO 


§  § 

s  I 


*8.1 

eg        « 


i  „   1 


8  j! 


R.      5 

00  VO 

rt          « 


0 

>-i 
ON 
w 
oo 


Q  0 

VO  VO 

N  ro 


§> 


$ 


I  «>  a" 

fi     F.    £ 


5    S 
H   " 


SN 


ON 


•s  J 

•o   ° 

.Is 


3J,  JO  - 


ro 


to  co  ^* 


574 


APPLIED   MECHANICS. 


•juiof  jo  XDiwpyjg 


I*. 

J3  Q 


8     II 
•B   e  <  JS 


•I  a -3  a 

ris 


t^» 

g 


II 

<u  c 

2  § 

w  A 

.S.S 


s| 

CA     & 


.S.S 


to  a, 


RIVETED  JOINTS. 


575 


•juiof  jo  Aouaiotyg 


£ 

I 
.s 

1-6 


C3 


I  II 


t* 


8 


iuv 


1 


l 


1 


•»»x  jo  -ON 


& 


jS'o 

<u  ^- 

•E 


© 
© 


4» 


N         00 
f)         CI 


O  N          VO 


O  O         O 

ID  r^       vo 

oo  f^        r^ 


#    ^      vS     3" 

eo      to          vo      vo 


eg    £       S 

•"T       r~*  el 

00        00  O* 


"  1 

I 


as 


R 

C?  Jo        ff 

••*•         »o     10 


3; 


s 


VO       vo 

vo         £ 
VO         VO 


VO      vo 
t^      r^ 

10         10 


en  en         en 


-"     -" 


«-0   «-a  ^13  SJ 

>  oj  >  «j  .C  4>  .5 

•C^'C^  J^-g^-0 

•  »-"    CU-i-i    CU  V)    OH  v;    OH       •£*  ^O  **^  ^O          c/3  *X?    c/}  *^3 

.S.S.2.S  .S.2.2.S     .2.2.2.2     .2.2.2.2 


576 


APPLIED   MECHANICS. 


«? 


By 

§ 


is 


S  S 


M  1 

f 


In  "U         c«  "O 

c  d       c  c 


CO  TJ 
C   C 


II 

d  c 


ro       ?O 


RIVETED  JOINTS. 


577 


•jinof  jo  Aouapiya 


ro 

.8 


-3, 


% 

'S-s 
I 

in 


Si 


1 


00 

1 


§     J  a 

•2   e  <  ^« 

8     O     $   PH 


IPJ'1 


13 


.S  d, 

c  c 


.S   d, 
C    G 


•JS3X  JO 


©I         j, 


r^. 
vS 


578 


APPLIED  MECHANICS. 


•»mof  jo  totwiDiya 


C/D 


2 

II' 


Il 

fO  ro 


S3 


Si 


« 


i      l-s 


10  vo 


VO  vo 

VO  vo 


vo 

vo 
r>. 

vo 
vo 


•-  - 

S  "2    g 


PS      2  3 
.£  &,    .fa  eu 


.§§. 


-C-a 

^  JB 


SL'T 


•»s»x  jo  -OK 


RIVETED  JOINTS. 


579 


•jtnof  jo  Aairapiga 


§  g 

I ! 

c/5 


jj 

§  a  s 


I 


1 


•JS9J.  JO  '0N 


> 


oo 


=  •8 
II 


II 


.s.s    .s.s 


II 

.s.s 


, 


•ft 


I! 

to      co 


2.     ° 
2s    3 


v£ 


580 


APPLIED   MECHANICS. 


•juiof  jo  Xouapiyg; 


O        ro 


Q* 
vo 


§  £ 


- 


ft 


« 


o  o 
r^  N 
ON  N 
NO 

IT)  LO 


£  a 

<  .2 


S»    8.  8  2 

t^        «S  t^  if) 

MM  M  00 

^-        ^-  1O  u-) 


j 

$  a, 


o        o 

00  O 

N  ^o 


OO 

W 


•jsaj,  jo  -oN 


•ft         c/> 

"o    .  'o 


0305  o  3  o  5 

.^H        Q-)*^4        P  t  .  ^         P^.^H        Q- 

.5.S.S.S  .S.S.S.5 

J:«H-^S***  '  -£  '  -fc 


vo 
\O 


•«!-  Tf 

to       m 
m      in 


00 

i-        VO 


0 
00 
if) 


LTI        O 

Ti- 
N  t^ 

vo       Tf 
»o       Co 


Q  O 

VO  Tf 

1-1  00 

Tf  ro 

CO  CO 


o  3  o  |     HI! 

cccc       cede 


vO 

-O 


O 
10 
VO 

co 
vo 


.?  -2 


RIVETED  JOINTS. 


•juiof  jo  Xouapiyg 

N                                      ON 

t^.                          O 
vO                           l^. 

"d            I 

r^.                       t^ 

rJ 

*?        „ 

8.            |  -o  | 

8            S 
$            % 

IO                             IT) 

O                                         Tf 

CO                              VO 

m                      co 

a 

'      .£                  ,             68            • 

ti  jj        9<  6    5  «£  •- 
1  e       J  '5  *  jj  2 

8              &' 

^             s 

OO                             00 

o>                      >o 
Sot 
£•% 

8 
1 

-i  |      1^41 

rt  cr,          S   o       S 

i:               ^         w  C. 

1O                          Q 
V?                        0 

0                           ^f 

£        S 

0 

_                              rt 

!i      1§ll 

CO                           OO                           ^O                             O                                ("O 

__-       Z 
£  "5     r  « 

fi   S  "~  t/3    c 

o                  o 

o                  o 

R 

ro 

582 


APPLIED  MECHANICS. 


vo 

Co 


& 


c    «> 
o    c 


8,       2 

10  o 


J'! 


^f  t-»  N 

^8-     ft     S> 


•c 

§     S 


<8 


o 

CO 


I  IS?! 


1 


&,  .s  &,  .a 
cede 


•3}vj(f  uo^j 


-witf  uojf 


S  S 

42-0  2  p 

II  I? 

§=  |  = 

.S 13  .ST3 

d  d  d  d 


•»9»X  JO 


RIVETED  JOINTS. 


583 


•?uiof  jo  XonaiojBg 


a    °s, 

t>»  NO 


S       « 

fO  ro 


f 


> 

tx  00 


a 


R 


ma 


lO 


ft 


CC         C   G 


'.§  I  .§  I 

.S.S     .S.S 


II  II 

.S.S     .S.S 

•3}VJ<f  py}$ 


APPLIED   MECHAA'ICS. 


•JUIOf  JO  ADU3IOIJJ3 


9 

•* 


1^  ON 


x;    ^ 

IJT 

w 


a 


^  S 

Is 


\ 

if 
CO 


O  O 

<»  £ 

oo.          en 


0 

00 
CO 


J8  §"g 

~   C/3     C 

C       ^     HH 


•7  ^> 

icjx.  O 


o  a  o 

5  ->5 

S  C  ^3 

'I  .g^ 


i 

•a 

I 


ii 


24 


©it! 


•JS9X  JO  'ON 


K1YETED  JO J NTS. 


585 


•juiof  jo  Xouapiya 

s, 

.5 


ss 


•s  Si 


§  SS 


O 
0\ 


ro  fO 


&     » 

I^>  OO 

vO  vO 


u->         oo 

I       S 


oq  q\ 

£         §8 


to          m 

04  t^ 

5  $ 


CO  CO 


CO  ro 


en  en 


«'C         «'C 

en  T3        en  'O 

C  d        d  d 


v;z%/  7^5- 


'V.'T  jo 


ill 


Q 
© 


© 


© 


586 


APPLIED  MECHANICS. 


8, 
I 

h 

§  s 

2    3 


8 


O 
vr> 

CSO^ 


8     S      ^   eg 
5      ff        J?     ft 

CO         CO  CO        CO 


vS 


O         to 

CO      vo 

ON      ON 


J* 


a 


o 

CO 


fill        I 


»O 
VO 

^t1 


JjJ  B|a|  C«| 


•E|-S|   •£;§•£ 3     2S 


c  c 


£«S« 

'C  43   C^3 


•8 


w 


00 

ft 


RIVETED  JOINTS. 


587 


•jmof  jo  Aatrapgjg 


4 

i 


1S9.I  JO  '0N 


i 


in  c/i 


g-g        §§ 

.b  a    .bo. 


O  Q 

00  \O 

M  »n 


C  g  S3    rj 

P  s  2s 

.53  ex  .be, 

c  c  d  d 


V?  V 


«o 

fO 

vo 


6 


So  «R 

£     R 


t^ 

in 


§c  e  c 

.3  O   g 

.be*  .b  a 

.S.S  .S.S 

ajfO  M  i^io  ^ 


'WWfucuj v;z?//  ;^;5« 


588 


APPLIED   MECHANICS. 


•juiof  jo  Xou»iogj3 


s, 


II 

I* 

C/D 


« 

II  II 


»^ 

vo 
fO 


o          o 


I 


1 


in  »n 

CO  CO 

1  1 


.5  2-S 


c  c 

'T£ 


a     . 


«  c-^ 

SM 

.S.S 


-b 


i- 


V18? 


© 


•»$»£  jo  -ON 


r?) 


RIVETED  JOINTS. 


589 


•juiof  jo  Xouapigg 


a"  . 


•i  «< 

0    £  PM 


a  tf-3  2"3 

,8    §    a  ?  J 


91 


in 


<     >f  .v 

^ 


R       3- 
3"       5? 

ro  S 


J     I 

m  m 


os-g 


c« 


1 


<o 


C    C 


00 


Si 


.M       Pi 

c  c 


590 


APPLIED  MECHANICS. 


•juiof  jo  AouaiDtga 


O 

\o 


3- 

10 


O  O 

N  fx 


O  O 

N  *r\ 

^  ^. 

N  N 


Q  Q 

S     « 

rj-  rf 


of  Rivets 
es. 


§§    §§ 

.h  tx    .h  a 


cd 


*7  j» 


en  t« 


cddc 
'r>    "'"t, 

it6  w   "i8  M 

'9ivtf  199}$ 


"  v  v 


RIVETED  JOINTS. 


591 


•juiof 


O         0 
r^       ON 


J2 


in 

w  rt  o 

« 


* 


O          0          0 
HI        r^       O 


oooooo 

rOi-i         M         >-<         r^Tj- 


$ 


§•!£§> 

fO  M  HI  Q 

t^      VO       VO       xO 


§000 
0         \rt        ^ 

CO        O         «5          M 


O          Id  M  CO        00 

M  IO          IT)  ID          VO 


ro^'-o          \o<-oroOfO^ 


- 

«§£">§ 

21 


w         1-1        \O 


O 
^ 


12 
£ 

3 
•oj5 

•S3 


oooo  oooooooo 

03030303    0303030303030303 


OOOOOOOO 


•jsax  jo  -OM 


fOVO        r^ 


592 


APPLIED   MECHANICS. 


•juiof 


la 


Ills 

MS-S! 


tr>       M 

ON       C\ 
10         vo 


o  o  o  o  o  o  o 

O»  O\  Oi  r^  "^r  c^  r** 

co  O*  co  O  N  O  ro 

o  a\  a\  <->  _  M  o 

^-  co  o»  co  co  ro  M 


VO  W 


\O       VO         TJ-       rj- 


o  o  o 

*tf-  M  M 

o^  «  N 

t^  IT)  n 

a\  a\  \o 


OOOO 
ir>        CO        >-i         •^> 

^roivooo 


oooooo 
t^M       c»       MOO       o» 


Q         Q         O  O         O 

9,    8    <£       ct   ^ 

t— i         ro       ^*  CO        t^* 


-83J    . 
1P1 


O  O  "~> 

ro  ro  M 

to  ro  w 

to  ro  r^ 

vo  u-,  10 


O        o        io>-n 

CO  CO  H-l  >-< 


M          >-i         I-H 


o       o       o       o 


(U  O  OJ 


<uo;<u<U 

<L)0)<U<L) 


.S     .S     .£     .S 

COJOO  KpO  r-pl  «|» 


lit 
co*C 


ill 


o      •- 

5  5 


£  & 


RIVETED  JOINTS. 


593 


GOVERNMENT   TESTS   OF  GROOVED   SPECIMENS. 


Tensile  Tests  of  .J-in. 
Grooved  Specimens 
Wroughl-Iron 

Punched. 

|jj 

"o 

P 

|2 

1| 

I  Is 

£ 

H 

£3 

Inch. 

Inch. 

0.48 

0.240 

48090 

0.46 

0.235 

46940 

0.46 

0.241 

49280 

0.49 

0.240 

55340 

0.44 

0.239 

51520 

0.47 

0.241 

49910 

0.97 

0.247 

49540 

0.98 

0.247 

49960 

0.94 

0.249 

50128 

0.96 

0.248 

46900 

0.98 

0.250 

46980 

0.96 

0.251 

46350 

i-47 

0.250 

37636 

1.50 

0.252 

37326 

1.48 

0.249 

41030 

1.48 

0.247 

39480 

i-47 

0.250 

37446 

i-45 

0.251 

39533 

1.96 

0.281 

43194 

i-95 

0.274 
0.282 

47499 
41360 

1.92 

0.279 

43080 

2.03 

0.250 

41140 

1.99 

0.248 

39575 

2.42 

0.280 

'36210 

2.40 

0.245 

42245 

2.47 

0.243 

42233 

2.46 

0.285 

42712 

2.48 

0.245 

38125 

2-44 

0.248 

41620 

2.97 

0.247 

38964 

2.98 

0.241 

41540 

2.96 

0.241 

39972 

2.92 

0.240 

41712 

2.98 

0.250 

40430 

3.95 

0.247 

40850 

Tensile  Tests  of  $-in. 
Grooved  Specimens 
Wrought-Iron 
Drilled. 

S 

^JZ 

o 

"bJD  ^ 

c  c 

M  u 

*0 

g 

3§ 

3 

'wof  j 

|0 

Jl 

i  v~~ 

£  ° 

H 

i5 

Inch. 

Inch. 

0.51 

0.249 

55787 

0.52 

0.245 

55905 

0.52 

0.275 

57480 

0.52 

0.276 

56000 

0.49 

0.248 

49600 

0.50 

0.248 

56700 

0.47 

0.275 

54880 

0.51 

0.276 

57800 

I.OO 

0.276 

54300 

1.02 

0.273 

57700 

I.OO 

0.276 

53800 

I.OO 

0.280 

52430 

I.OO 

0.252 

49400 

1.02 

0.275 

54060 

I.OI 

0.247 

52770 

I.OO 

0.278 

54600 

1.50 

0.276 

49J3° 

1-52 

0.273 

51300 

1.48 

0.251 

47220 

1.51 

0.273 

53400 

1-52 

0.275 

54l8o 

1.50 

o  276 

54600 

1.48 

0.274 

56250 

1.50 

0.249 

46260 

2.OI 

0.275 

459oo 

2.05 

0.279 

46820 

2.OO 

0.275 

47950 

2.00 

0.278 

49640 

2.00 

0.286 

44650 

2.00 

0.275 

50780 

2.02 

0.279 

48850 

2.OO 

0.277 

49840 

2-51 

0.244 

44980 

2.52 

0.280 

40150 

2.51 

0.282 

43150 

2.50 

0.244 

455oo 

2.51 

0.285 

46500 

2.49 

0.242 

49520 

2.49 

0.242 

— 

2.50 

0.280 

44780 

3-02 

0.250 

45700 

3.02 

0.249 

44870 

3.00 

0.240 

46760 

3.00 

0.250 

45700 

2.93 

0.242 

47950 

0.250 

48740 

2.98 

0.279 

459°° 

3.01 

0.281 

44410 

Tensile  Tests  of  £-in. 
Grooved  Specimens 

Steel  Plate 

Punched. 

S 

.c    •> 

o 

s! 

-1 

1 

&  cr 

•5^ 

H 

P 

Inch. 
0.49 

Inch. 
0.250 

65120 

0.47 

0.249 

67010 

0.48 

0.249 

63420 

0.48 

0.248 

66550 

0.48 

0.247 

67060 

0.47 

o  248 

65300 

o-99 

0.249 

59840 

I.OO 

0.250 

62160 

I.OI 

0.249 

68246 

0.96 
0.96 

0.250 
0.248 

67330 
65966 

0.95 

0.245 

62700 

i-45 

0.248 

64080 

1-45 

0.252 

64000 

i-45 

0.249 

61025 

i«5* 

o  251 

59420 

1.96 

i-93 

0.250 
0.252 

599oo 
63500 

1.98 
1.96 

0.250 
0.251 

59350 
59060 

2-49 

0.249 

58100 

2.47 

0.249 

63900 

2-43 

0.250 

61640 

2-95 

0.251 

56530 

3-oi 

0.249 

58780 

3-04 

0-253 

555oo 

2.97 

0.252 

60060 

2.98 

0.251 

54050 

2.97 

0.249 

56040 

Tensile  Tests  of  J-in. 
Grooved  Specimens 
Steel  Plate 

Drilled. 

| 

si  o 

C   C 

W  ** 

^o 

£*" 

$ 

C/5     ,y 

v    • 

^  f-     t/J 

4=0 

II 

E  "-  "~ 

I*8 

-Mia 

* 

P 

u> 

Inch. 

Inch. 

0.52 

o-54 

0.246 
0.248 

67890 

67160 

0-53 

0.247 

66870 

0.50 

0.247 

65610 

0.51 

0.249 

66370 

0.51 

0.250 

67420 

0.52 

0.248 

67750 

0.52 

0.252 

61910 

1.03 

0.247 

57090 

1.02 

0.250 

66390 

1.02 

0.246 

66770 

1.02* 

0.250 

67730 

I.OI 

0.247 

66020 

I.OO 

0.251 

67010 

I.OO 

0.247 

64450 

I.OI 

0.250 

66090 

i-54 

0.250 

64390 

1.52     0.251 

63350 

1.5° 

0.253 

64370 

i-54 

0.248 

64895 

2.O2 

0.252 

64320 

2.00. 

0.251 

62970 

2.00 

0.251 

60910 

2.50 

0.248 

59260 

2.50 

0.252 

63250 

2-53 

0.248 

59390 

3-03 

0.251 

61577 

3.00 

0.249 

59080 

3-02 

0.251 

59550 

3-02 

0.250 

59700 

3.00 

o  250 

63370 

3.00 

0.251 

58630 

3-03 

0.252 

63940 

594 


APPLIED   MECHANICS. 


IRON  POUCHED. 


IRON  DRILLED. 


STEEL  PUNCHED. 


STEEL  DRILLED. 


Tensile  Tests  of 
Grooved  Wrought- 
Iron  Plates. 

Tensile  Tests  of 
Grooved  Wrought- 
Iron  Plates. 

Tensile  Tests 
of 
Grooved  Steel  Plates. 

Tensile  Tests 
of 
Grooved  Steel  Plates. 

% 

•BM 

c'c 

£~ 

^  6- 

JJC/JJ 

1 

•a* 

bJ3  o 
|J 

w  sr 

WO!   «' 

i 

If 

Jjl-H 

^  cr 

WC/2    ui 

, 

•SJ 

c"e 
U1"1 

OTrJ, 
JjdTjj 

*2 

? 

J 

ls.s 

p 

£ 

.a 

B 

IS.fi 

1 

£ 

•lS.a 

t> 

1 

g 

ib 

5 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

Inch. 

I.OI 

0-373 

47000 

0.98 

0.376 

50870 

1.99 

0-365 

61890 

1.97 

0.369 

63620 

0.98 

0.370 

47520 

0.98 

0-377 

52660 

0.99 

0-494 

70080 

I.OO 

0.498 

66220 

2.OO 

0.382 

39760 

1.98 

0-379 

49710 

I.OO 

0.492 

68130 

o-99 

0-495 

66800 

2.  02 

0.383 

36630 

2.OO 

0.380 

49830 

1.50 

0.497 

66340 

I.OO 

0.500 

67000 

2-39 

0.390 

37600 

2.50 

0.390 

50250 

i-5i 

0-494 

63810 

i-53 

0-497 

65930 

2.98 

o-395 

36340 

3-00 

0.392 

45I5o 

1.99 

0.499 

55930 

1.50 

0.498 

66270 

2.98 

0.392 

39210 

3-00 

0-393 

47540 

1.97 

0.500 

64260 

1.98 

0.504 

67510 

3-47 

0.390 

37680 

3-50 

0.392 

43940 

2-43 

0.502 

52050 

2.03 

0.502 

66730 

3-47 

0.389 

38340 

3-49 

0.390 

46490 

2-51 

0.504 

64360 

2.50 

0.497 

67950 

0.97 

0.467 

-   50820 

0.99 

0.477 

47140 

3-00 

0.503 

60320 

2.52 

0.501 

67440 

1.48 

0.506 

45090 

1.  00 

0.479 

48370 

2-99 

0.503 

62430 

3-oi 

0.502 

66310 

1.49 

0.506 

45050 

1.49 

0.510 

51240 

3-50 

0.503 

49430 

3-oi 

0.503 

66190 

1.91 

0.513 

42500 

1.49 

0.512 

51510 

3-50 

0.505 

48270 

3-49 

0.504 

64920 

1.97 

0.512 

43430 

1.98 

0.514 

50050 

4.00 

0.497 

48010 

3-50 

0.502 

65210 

2.47 

0.516 

39410 

1.98 

0.516 

47790 

4.00 

0.499 

55190 

3-99 

0-499 

64470 

2.41 

0.513 

39720 

2.51 

0.520 

4558o 

3-99 

0.501 

55780 

4.00 

0.498 

64810 

3.00 

0-515 

38950 

2.52 

0.516 

44960 

3-99 

0.498 

46250 

4.00 

0.503 

64690 

2.90 

0.517 

37290 

3-oo 

0-5*5 

44980 

I.OI 

0.613 

66720 

4.00 

0.498 

64140 

3-5° 

0.520 

37800 

3.01 

0.519 

4703° 

1-52 

0.612 

64800 

o-99 

0.619 

60290 

3-49 

0-513 

37770 

3-51 

o-5I3 

46170 

i-5° 

0.615 

64400 

1.49 

0.614 

63610 

4.00 

0.515 

35730 

3-49 

0.514 

44760 

2.50 

0.618 

58060 

1.49 

0.616 

63450 

4-03 

0.516 

36690 

3-99 

0.510 

4533° 

2.52 

0.619 

58780 

2-49 

0.620 

59170 

3-99 

0.511 

37000 

3.98 

°-5I3 

45000 

2.99  '  0.617 

57180 

2.50 

0.619 

59600 

4.03 

0.508 

37420 

4.00 

0.506 

46100 

3-46 

0.615 

58410 

3-oi 

0.617 

59270 

0.97 

0.614 

49770 

o-97 

0.628 

47220 

3-51 

0.615 

57190 

3-50 

0.614 

61610 

I.OI 

0.619 

52960 

1.  00 

0.626 

4835° 

4.04 

0.612 

54450 

3-49 

0.617 

62060 

1.48 

0.618 

46320 

1.52 

0.625 

47170 

4-°3 

0.614 

57380 

4.00 

0.615 

60330 

i.S2. 

0.620 

46750 

1.49 

0.629 

4653° 

I.OI 

0.721 

67930 

4.01 

0.617 

61120 

2.99 

0.614 

40140 

2.98 

0.613 

48220 

I.OO 

0.718 

67620 

0.96     0.726 

58480 

3-5° 

0.615 

37480 

3.46 

0.616 

4777° 

1.50 

0.719 

62890 

I.OI 

0.727 

58790 

3-5° 

0.616 

36940 

3-47 

0.617 

449°° 

3-50 

0-735 

56730 

i-5* 

0.726 

59290 

4.04 

0.619 

37310 

3-91 

0.625 

44840 

3-51 

0-733 

54220 

3-5o 

0.736 

58700 

;  0.98 

0.678 

50840 

3.96 

0.626 

45ioo 

3-49 

0.729 

59x8o 

] 

I.OI 

0.682 

46590 

0-99 

0.695 

47500 

i-49 

0.688 

4597° 

0.99 

0.691 

52780 

3.48 

0.691 

4035° 

I-5I 

0.692 

48470 

(  3-53 

0.692 

39380 

3-44 

0.700 

47750 

3-49 

0.692 

46350 

L 

TENSILE    TESTS  OF  RIVETED  JOINTS.  595 


Next  will  be  given  the  two  series  of  tests  already  referred, 
to,  with  Mr.  Howard's  analysis  of  them. 

TENSILE    TESTS    OF    RIVETED    JOINTS. 

"  Earlier  experiments  on  this  subject  made  with  single  and 
double  riveted  lap  and  butt  joints  in  different  thicknesses  of 
iron  and  steel  plate,  together  with  the  tests  of  specimens  pre- 
pared to  illustrate  the  strength  of  constituent  parts  of  joints, 
are  recorded  in  the  report  of  tests  for  1882  and  1883. 

From  the  results  thus  obtained  it  appeared  desirable  to 
institute  a  synthetical  series  of  tests,  beginning  with  the  most 
elementary  forms  of  joints  in  which  the  stresses  are  found  in 
their  least  complicated  state.  To  meet  these  conditions,  a 
series  of  joints  have  been  prepared  which  may  be  designated  as 
single-riveted  butt-joints,  in  which  the  covers  are  extended  so 
as  to  be  grasped  in  the  testing-machine  ;  thereby  enabling  one 
plate  of  the  joint  to  be  dispensed  with,  and  securing  the  test  of 
one  line  of  riveting. 

Such  a  joint,  made  with  carefully  annealed  mild  steel  plate 
of  superior  quality,  with  drilled  holes,  seems  well  adapted  to 
demonstrate  the  influence  on  the  tensile  strength  of  the  metal 
taken  across  the  line  of  riveting,  of  variations  in  the  width  of 
the  net  section  between  rivets,  and  variations  in  the  compres- 
sion stress  on  the  bearing-surface  of  the  rivets  ;  elements  which 
are  believed  to  be  fundamental  in  all  riveted  construction. 

This  series  comprises  2 16  specimen  joints,  the  thickness  of  the 
plate  ranging  from  J"  to  |7/,  advancing  by  eighths.  The  covers 
are  from  y3^-"  to  yV'-  The  rivets  are  wrought-iron,  and  range  from 
-3-$"  to  lyV  diameter;  they  are  machine-driven  in  drilled  holes 
iV'  lai"ger  in  diameter  than  the  nominal  size  of  the  rivets.  Ten- 
sile tests  of  the  material  accompany  the  tests  of  the  joints. 

From  each  sheet  of  steel  two  test-strips  were  sheared,  one 
lengthwise  and  one  crosswise.  The  strips  were  2\"  wide  and  24" 
to  36"  long;  they  were  annealed  with  the  specimen  plates, and  had 
their  edges  planed,  reducing  their  widths  to  i|"  before  testing. 


596  APPLIED  MECHANICS. 

Micrometer  readings  were  taken  in  10"  along  the  middle  of 
the  length  of  each. 

The  strength  and  ductility  appear  to  be  substantially  the 
same  in  each  direction.  But  the  practice  of  the  rolling-mill 
where  these  sheets  were  rolled  is  such  that  nearly  the  same 
amount  of  work  may  have  been  given  the  steel  in  each  direc- 
tion ;  that  is,  lengthwise  and  crosswise  the  finished  sheet. 

The  ingots  of  open-hearth  metal  are  first  rolled  down  to 
slabs  about  6"  thick,  then  reheated  and  rolled  either  length- 
wise or  crosswise  their  former  direction,  as  best  suits  the  re- 
quired finished  dimensions. 

The  tensile  tests  show  among  the  thinner  plates  a  relatively 
high  elastic  limit  as  compared  with  the  tensile  strength  ;  in  the 
•f$'f  plate  the  percentage  is  72.2,  while  with  the  f"  plate  the 
percentage  is  found  to  be  53.3. 

It  is  noticeable  that  the  thinner  plates  particularly  exhibit 
a  large  stretch  immediately  following  the  elastic  limit,  and  the 
stretching  is  continued  at  times  under  a  load  lower  than  that 
which  has  been  previously  sustained.  It  is  characteristic  of  all 
the  thicknesses  that  a  considerable  stretch  takes  place  under 
loads  approaching  the  tensile  strength — in  some  cases  the 
stretch  increases  5  to  6  per  cent,  while  the  stress  advances  1000 
pounds  per  square  inch  or  less.  Herein  is  found  a  valuable 
property  of  this  metal  as  a  material  for  riveted  construction. 
The  stress  from  the  bearing-surface  of  the  rivets  is  distributed 
over  the  net  section  of  plate  between  the  rivets,  due  to  the  large 
stretch  of  the  metal,  with  little  elevation  of  the  stress,  and  a 
nearer  approximation  of  uniform  stress  in  this  section  attained 
than  is  found  in  a  brittle  or  less  ductile  metal.  The  joints  were 
held  for  testing  in  the  hydraulic  jaws  of  the  testing-machine, 
having  24''  exposure  between  them.  A  loose  piece  of  steel 
the  same  thickness  as  the  plate  was  placed  between  the  covers 
to  receive  the  grip  of  the  jaws,  and  avoid  bending  the  covers. 

Elongations  were  measured  in  a  gauged  length  of  5",  the 
micrometer  covering  the  joint  at  the  middle  of  its  width.  Loads 


TENSILE    TESTS  OF  RIVETED  JOINTS.  597 

were  applied  in  increments  of  1000  pounds  per  square  inch  of  the 
gross  section  of  the  plate,  the  effect  of  each  increment  determined 
by  the  micrometer,  and  permanent  sets  observed  at  intervals. 

The  progress  of  the  test  of  a  joint  is  generally  marked 
by  three  well-defined  periods.  In  the  first  period  greatest 
rigidity  is  found,  and  it  is  thought  that  the  joint  is  now  held 
entirely  by  the  friction  of  the  rivet-heads,  and  the  movement  of 
the  joint  is  principally  that  due  to  the  elasticity  of  the  metal. 

The  second  period  is  distinguished  by  a  rapid  increase  in 
the  stretch  of  the  joint ;  attributed  to  the  overcoming  of  the 
friction  under  the  rivet-heads  and  closing  up  any  clearance 
about  the  rivets,  bringing  them  into  bearing  condition  against 
the  fronts  of  the  rivet-holes.  Rivets  which  are  said  to  fill  the 
holes  can  hardly  do  so  completely,  on  account  of  the  contrac- 
tion of  the  metal  of  the  rivet  from  a  higher  temperature  than 
that  of  the  plate,  after  the  rivet  is  driven. 

After  a  brief  interval  the  movement  of  the  joint  is  retarded, 
and  the  third  period  is  reached.  The  stretch  of  the  joint  is 
now  believed  to  be  due  to  the  distortion  of  the  rivet-holes  and 
the  rivets  themselves.  The  movement  begins  slowly,  and  so 
continues  till  the  elastic  limit  of  the  metal  about  the  rivet-holes 
is  passed,  and  general  flow  takes  place  over  the  entire  cross-sec- 
tion, and  rupture  is  reached.  These  stages  in  the  test  of  a  joint 
are  well  defined,  except  when  the  plates  are  in  a  warped  condi- 
tion  initially,  when  abnormal  micrometer  readings  are  observed. 

The  difference  in  behavior  of  a  joint  and  the  solid  metal 
suggests  the  propriety  of  arranging  tension  joints  in  boiler  con- 
struction and  elsewhere  as  nearly  in  line  as  practicable. 

The  efficiencies  of  the  joints  are  computed  on  the  basis  of 
the  tensile  strength  of  the  lengthwise  strips,  this  being  the 
direction  in  which  the  metal  of  the  joints  is  strained.  The 
efficiencies  here  found  are  undoubtedly  lowered  somewhat  by 
the  contraction  in  width  of  the  specimens,  causing  in  most  cases 
fractures  to  begin  at  the  edges  and  extend  towards  the  middle 
of  the  joint.  Of  the  entire  series,  88  joints  have  been  tested ; 
tfr*  2",  £ ",  and  f "  plates  yet  remain." 


APPLIED  MECHANICS. 


w  o 


w 


o 
3 
u 

U 

a 

rt 

1 

.a 

| 

rt 
C 

1 
fl               .1 

,  w        .              e  <" 

>»i    ^ 

;3  ri    .:S    .    .       S  rt    .    . 
'55  ~~  o'ir  °  °        M~  °  ° 
"xQ.yQQ      >,>«QQ>, 

gjd           g                       ^^                JSd 

Do. 

ne  silky. 
Iky,  lamellar.  , 
ne  silky, 
ne  silky,  slight  lamination. 
Iky.  lamellar. 
Do. 
me  silky. 
Do. 
Do. 
Do. 

Iky,  slight  lamination, 
ne  silky,  surface  blister. 
Iky,  slight  lamination. 
Iky,  stratified. 
Iky. 
ne  sil'ky. 
Ikv,  lamellar. 
Iky.  slight  lamination. 
Iky,  laminated,  stratified. 
Iky,  laminated. 

fe  j/5    fc           c/5  j/5        c/5 

t,  c/)  fc  fe  i/)     tt. 

(73ttic75c7}c7:)fc<i/3c/3c/;i75 

I       (  c 

C'Ss.,  jj 

is     rt 

pE?R«°     i^li 

«  -*oo  »ooo  10  10  rooo  O  o 

2^^U°  o,^ 

Ssc  2| 

S^r.o-00.        -^»«- 

S'^-S 

d 

^6     J 
•S  Sc 

5                  § 

8s 

1 

1 

W  "              u     • 

«lv§l|p;l>    &^H 

OOOOQOOOOQO 
•^•NO  OQNO    o    lOr^hx^-O    f^ 

C 

rt 
u 

£               ^ 

Tf                                                         Tf 

00 

10 

o 

1 

*~      && 

OOOOOQ         COOOO 
•oor^Q^ONO       Not^io-^--^- 

ooooooooooo 

i/>  10NO  NONOO"M^-iO->J- 

t^-OO    iO  C     O     0     u-)  t^.  C>OO 

^^'S 

ilj 

•^   t^  t-^oo  t^  t^  10       r^co  ONOO  NO 

•CNlMNtNIPllN          NCMININCNl 

l&SSs-H^&SS 

54feg6g4S«S 

1/3  -^  rt 

w 

«     «  s 

c       )S  <u 

•§  00  OO    ONOO  00    tv        00  OO    ONOO  OO 

NO    w    m  M    t-»  CNI    O    •*  •*  O  00 

H&5I??£!J 

— 

<|     -s 

8^^858'S     ^S^S; 

O-  ON  ONOO    ON  ON  ON  O>  ON  ON  O 

S;^o3  s;  R  SN^  ^°£ 

•5        ^ 

•g  

T  .    .  T  "? 

pE 

c 
o 

"3        C 

I                  | 

"5 

jj 

|°6i 

•S  d  d  d  d  d      ^  d  d  d  d 
bf'o  T3-a*UT3      tfl-a-o-a-a 

iJ                        U 

•5  6  d  o  6  6  6  6  d  o  d 

|™^]^ 

^  d  d  d  d  d  d  d  d  6 

y 

s~-^  «' 

jl 

o  c  -n  w 

|-B«CH»««|5"R-B    <«£«S«S«S 

|| 

«OOQ«  1    <«u^ 

fcos~-*JSZ~ 

fcoS-^JS2 

l~2 

i§ii§?  s?«? 

M    <N|    r«  Tf  IONO    t^OO    ON  O    H 

W   ro  ^t-  iovo   txoo  ON  O   w 

TABULATION  OF  0.    H.    STEEL    STRIPS. 


599 


c      d 
.2     .2 

rt       cS 

C        C 

a  .s 

rt  53  .2 

^Ij^ 
tag  bfi 

3  .2  So" 


c 
g     .2 


m 

r. 


edi 
t  l 
lla 

la 
lla 
t  l 


la 
fied 


£     £ 


i££££    ^ 

cflc/ir^wr^xx^3 


fc  te         c/j  (73  c/5  c/5 


C/)COCO  C/3CCW3       C/5 


w    H          OOp    ro  M    c>  TJ-VO    O 


o  in  ^-  r**vo        N  N  rf  H  ^t-       io\o  t-^ao  t^  t^  moo  -^-vo       tnvo  *OH\OVO\OIOOISS       •^•cM-^-oor^oi 

rOMCSMM  CSNMNIN  NNOISNNNMNN  NMMNWNNNMtM  NNNNNMrOl 


ooo   ooooooooo 


OOOOOOOO         OOOOOOOOOO         QOOOO 
t^ro*^-o>^t*t^rocn       t^inoofoi^NOco^       ooin-^-M 


-  ON  t^  rOOO    - 
mininm 


IN  H  00     O  00 

r~>       in  f-oo  in 


m\o  M  c>  i 

>•&     %$££M 


OOOO         OOOOO         OO1 
•^•oo  wvo        o*  in  O  M  *o       cow1 

»  tx  ro  0)  vO    VO  •<*•  -^-vo  10    fO  TJ- 


O    OOOOOOOOOO    OOOOOOOO 

,  IT)          H     M    lOOO  VO    C*    O  <N     rO  tx          O  U~)VO    LOCO    N  CO    IO 


t^  i-^vo  TJ-  10  u*>  LO       loco  vo  vo  tx  i--  -^f-  10  LOVO        miomM  w  ^J-CON 


oooo     -Sooooooooo  ?  ooooooooo     £0000000 

T3  T3  T3  *o      tiro  'O'O'a'a'a'O'O'O  {g'O-o'O'aTJ'a'a'D'o      bo^a  -o  *c  »o  -a  -o  -o 

c  o  *  c 

3  £  3 


invO    •<*•  N    ro 
«"  {7  f?  fT  N 


600 


APPLIED  MECHANICS. 


z 

Q 
W 

O 

pearance  of  fracture. 

c 
o 

w 
c                      c      c          c              v 

1       -I        'I  -s.-  -s     -i 

1          1             -   II   1         8 

S               -3                     ~      8  "*>      S           •  ° 

-:    ll    i-  1  II  1  ii| 

1       11     Jrlilf  Is  1  III 

o       O*QMCO          n^^r^:^       <«>       en      —tnO 

lamellar, 
slight  lamination, 
o. 

o. 
stratified, 
slight  lamination. 

0, 

< 

>i  >>Q  Q>i>.>»        4)XW(y          >.£>>>,        t^.^^ 
^Jsj             J^^J^      •'C-^CC        •*  -^  -^  -^        -^  -^  >- 

>,>M   Q  >,>,    >;>, 

Ji^                   J«!J<1       ^!^! 

s 

C/3t/3            C7)C/2C/3       bnc/jfefe       C/2C/3C/2C/2       C/2t//t> 

C/2C/)                  t/2{/5       C/3C/3 

W 

i§      « 

J 

•J 

Vj   r^  r^oo'  vd   4-  c^oo*         r^vo"  od   M        't^  M   «  >o       »o  u^oo' 

M   o   •<*•        M   iovo         •*  0 

0  g      rt 

a 

W 

^ 

o 

o-2  c'S 

ScTScTS^c?    S^cTcT    cT^^JT    Rff'g 

00  m  •*•      ao  t^  •*•       Tt-t^ 

ll 

'  «.s          ^ 

i  ^  M 

ON                                                        *5>                                         ^                                       f^ 

t^                m              ui 

c/)   « 

1   53  S          u 

H8*B 

^^   in  \T)  too  vo  in  10       in  mvo  in      om^nm       ininin 

III  MI  !f 

SH 

«n  ^ 

1            ^          s        1 

ro                  0                 m 

1     1*    H 

co  W 

H 

1  4i 

en 

1111  ill  II! 

!!!  HI  If 

J  U 
W  > 

»ls 

C   iovo   O   *  M   M    •*•        Hroc>ro       t)oo«io       i-   -^j-vo 

^"jr  as^?  ^5- 

e/>.2  J5 

1?    

^  ^  ^    °  °  H     [:  • 

£a 

..       |« 

^'ct^SS^!^^    i^Rcgo?    cg^cSocT     82^ 

JT£?    ^S?    ^? 

T 

"3.1       EC 

3  <fl          ^ 

o 

«.l             1 

JllillSS  IH?  HH  S?? 

1H  ?§a  11 

fan 

£ 

|H«W,HHMH         HMHM         M         M  H         MMM 

C 
O 

•J      c 

.2                                       |                       .2                       | 

«           8           „ 

0 

p 

l°* 

u                  J5           u           ^ 

^  d  d     •£  6  6      £  d 

en  T3  -O        &QT3  T3        Jg  -Q 

s       g       s 

U             J             U 

J 

S1315  en' 

^ 

03 

|-S2  c 

c 

H 

li 

^•JS^OCUOJ    f^tyiH^     pit/jH^     t>^^ 

>s*  >N  i  >N 

O  MM  tn 

S5»| 

O    M    CN    fO  -^-  IOVO           M    CM    CO  M*         t^OO    O\  O           IO\O    t^ 

ifi  if?  &i 

RIVET  METAL   FOR  RIVETED  JOINTS. 


601 


cT      o"          ^     oo*      o*      w*      "*£      oT      ON      -^ 

(NW                  Nt-*'-WC<WWW 

e?       J? 

c 

.2   » 

""d^H  *^'?  8-  !  !  t  *?^ 

*'i'  * 

y-y  ^?5"  ?  «"  li  ?  R  « 

•»'  ^  S^ 

a 
o 

1 

§ 

^Vo;\  5^  a  \  M  ^^ 

tx^  '  ocf^  *          W^«^O^O\         M^         C^          O      *    fO     »  d 

«  ^  •     •  vo" 

s 

^.-  ^-     ^.-  °2  '  ct  •  S  •  S?  •  J?  '  «   -VS  • 

-?"  « 

gjl 

£"0 

y-*M                Tj-^O-^-OOOOOOv 

*           M 

8 

**  V                   -i3 

•  H'M             N        o        mrom-frioin 

s   5 

23          * 

s*  

<  r5                 S   u 

-C    ^-         -^-                iO         u">        VO         ^         VO           t^.        OO         OO 

s-  s 

Q   V 

e 

is 

tJfOlO              OOrOlOMVONOOM 

UMW            •4-MOiooovdr-.oo' 

•*•        00 

11 

a 

11 

g.ff  ff     ^ff-yasjj'g.fi 

S    12 

•^  be             </)  ^ 

«i    §»      tis^&^SS^ 

—  'ITJIA             mmiou^ioioioio 

VO          CO 

1  s  § 

H  •*-*              "rt 
o 
H 

ill  |  HI!!  fl 

0         0 
vg         | 
10         10 

<y  . 

Q                          A        3     fl 

^•oo     S-       o?     m2^5-Rv2     ^;J 

00           O 

J                 CAM 

rt            5 
S             H 

oo           oooooooo 
•10^-            5^0-^^SooxoO 

2?C>ON                f>.^J-CI           N           rOONVO           tx 

^oooo             2"^§        §        c?c?&°o 

II 

ill 

.sf.f     s   1   1   Is  f  i  1 

10         10 

cT      o 

13  A    • 

3ifc 

33* 

•^•^              \O*OON^N           NVOVO 

E-     ^ 

111  si 

c-w   ^      •*   •«    <*»   •*    *   -   ^g   -s 

"R  "R 

M              M 

M 

#3* 

*  ?     1  f  &  S,  &  S  S  § 

•«  i;    | 

6O2 


APPLIED    MECHANICS. 


TABULATION    OF   SINGLE 

i"  STEEL 


No.  01 
Test. 

Sheet  Letters. 

Pitch. 

No.  of 
Rivets. 

Width 
of 
Joint. 

Nominal 
Thickness. 

Size  of 
Rivets 
and 
Holes. 

Actual 
Thick- 
ness   of 

Plate. 

Lap. 

Plate. 

Covers. 

Plate. 

Covers 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

1308 

F 

A 

A 

if 

6 

9-75 

i/4 

.242 

2 

1309 

F 

A 

A 

44 

6 

44 

44 

44 

.242 

2 

1310 

F 

A 

A 

it 

6 

10.50 

44 

44 

.242 

2 

13"         F 

A 

A 

11 

6 

4' 

44 

44 

.249 

2 

1312          F 

A 

A 

if 

6 

11.25 

44 

44 

.244 

2 

1313 

F 

A 

A 

" 

6 

" 

44 

" 

44 

-243 

2 

i3H 

F 

A 

A 

it 

6 

10.49 

44 

tt** 

.248 

2 

1315          F 

A 

A 

" 

6 

" 

44 

44 

.242 

2 

1316          F 

A          A 

i| 

6 

11.27 

tt 

" 

44 

.244 

2 

1317  |       F 

A          A 

44 

6 

44 

44 

44 

44 

.246 

2 

1318          F 

B     1     B 

2 

6 

12.01 

44 

it 

44 

•245 

2 

1319         G 

B          B 

" 

6 

" 

44 

" 

44 

.240 

2 

1320         G 

B 

B 

2* 

6 

12.76 

44 

" 

'4 

243 

2 

1321         G 

B 

B 

14 

6 

44 

44 

" 

44 

•243 

2 

1322         H 

D 

D 

2} 

6 

I3-5I 

44 

44 

tt 

•245 

2 

1323  :       H 

D 

C 

" 

6 

" 

44 

" 

tt 

•247 

2 

1324         F 

A 

A 

1* 

6 

11.26 

44 

" 

T§*  J 

.248 

2 

1325          F 

A 

A 

44 

6 

" 

44 

44 

" 

•245 

2 

1326         G 

B 

B 

2 

6 

12.  OO 

44 

" 

4' 

.241 

2 

1327         G 

B 

B 

44 

6 

44 

44 

44 

•• 

.242 

2 

1328  1       G 

B 

C 

2* 

6 

12.76 

" 

it 

tt 

.241 

2 

1329  !       G 

C 

C 

14 

6 

" 

44 

44 

44 

.242 

2 

'33° 

L 

C 

D 

2* 

6 

33-50 

44 

44 

44 

.248 

2 

I33t 

H 

C 

D 

44 

6 

44 

it 

" 

it 

.246 

2 

1332 

H 

D 

E 

2f 

6 

14.25 

u 

44 

.248 

2 

1333 

H 

D 

E 

" 

6 

44 

44 

44 

44 

.248 

2 

1334 

M 

E 

E 

a* 

6 

15.00 

ti 

ti 

44 

•  243 

2 

1335 

0 



.... 

" 

6 

" 

it 

44 

44 

•245 

2 

1336 

H 

C 

C 

2f 

5 

I3-I3 

<t 

it 

44 

238 

2 

1337 

L 

C 

C 

" 

5 

44 

44 

44 

it 

.252 

2 

1338 

G 

B 

B 

2 

6 

12.00 

44 

44 

tf  *i 

.238 

2 

J339 

F 

B 

B 

11 

6 

11 

44 

44 

44 

.248 

2 

1340 

G 

B 

C 

2* 

.  6 

12-75 

44 

ii 

" 

.240 

2 

i34i 

G 

C 

C 

11 

6 

44 

it 

it 

M 

.242 

2 

1342 

L 

C 

D 

2* 

6 

I3-5I 

44 

4i 

it 

.250 

2 

1343 

L 

D 

D 

" 

6 

" 

" 

it 

" 

.250 

2 

TABULATION   OF  RIVETED  JOINTS. 


603 


RIVETED    BUTT-JOINTS. 
PLATE. 


Sectional  Area 
of  Plate. 

Bearing 
Surface 
of 
Rivets. 

Shear- 
ing 
Area  of 
Rivets. 

Tensile 
Strength 
of  Plate 
per 
Sq.  In. 

Maximum  Stress  on  Joint  per  Sq.  In. 

Effi- 
ciency 
of 
Joint. 

Tension 
on  Gross 
Section  of 
Plate. 

Tension 
on  Net. 
Section  of 
Plate. 

Comp.  on 
Bearing 
Surface 
of  Rivets. 

Shear- 
ing of 
Rivets. 

Gross. 

Net. 

sq.  in. 

2.360 

sq.  in. 

1-452 

sq   in.     sq.  in. 
.908        3.682 

Ibs. 
61740 

Ibs. 
41690 

Ibs. 
67770 

Ibs. 
108370 

Ibs. 
26720 

67-5 

2.360 

1.452 

.908  !    3.682 

61740 

42180 

68560 

109640 

27040 

68.3 

2    541 

1.634 

.907 

3.682 

61740 

42540 

66160 

119180  v 

29360 

68.9 

2.6l5 

1.681 

•934 

3.682 

61740 

43  1  7° 

67160 

119810 

30660 

69.9 

2-745 

1.830 

•  9*5 

3-682 

61740 

44920 

67380 

134750 

33490 

72.8 

2-739 

1.827 

.912 

3.682 

61740 

44520 

66750 

133720 

33120 

72.1 

2.602 

1.486 

i  .  116 

5.300 

61740 

40700 

71270 

94890 

19980 

65.9 

2.541 

1-452 

1.089 

5-300 

61740 

40000 

70000 

93330 

19180 

64.8 

2.750 

1-652 

1.098 

5.300 

61740 

40980 

68210 

102630 

21260 

66.4 

2.772 

1-665 

i  .  107 

5.300 

61740 

4I43° 

68980 

103750 

21670 

67.1 

2    942 

1.840 

I.  102 

5-300 

61740 

42180 

67450 

112610 

23420 

68.3 

2.882 

1.802 

I.oSo 

5.300 

62660 

43000 

68770 

"475° 

23380 

68.6 

3.100 

2.007 

1.093 

5.300 

62660 

43060 

66520 

122140 

25190 

68.7 

3.100 

2.007 

1.093 

5.300 

62660 

44030 

68000 

124880 

2575° 

70.3 

3-310 

2.207 

I.I03 

5.300 

59180 

43040 

64540 

129150 

26880 

72.7 

3-337 

2.225 

I  .  112 

5-3°° 

59180 

43810 

65700 

131460 

27580 

74-o 

2.792 

1.490 

I  .302 

7.216 

61740 

38650 

72480 

82870 

14950 

62.0 

2.756 

1.470 

1.286 

7.216 

61740 

39430 

73930 

84510 

15010 

63-8 

2.892 

1.627 

1.265 

7.216 

62660 

40340 

71700 

92210 

16170 

64-4 

2.909 

1-638 

I.27I 

7.216 

62660 

41280 

73310 

94480 

16640 

65-9 

3-075 

1.810 

1.265 

7.216 

62660 

42290 

71850 

102810 

18020 

67-5 

3.088 

1.817 

I.27I 

7.216 

62660 

42750 

72650 

103860 

18290 

68.2 

3.343 

2.046 

1.302 

7.216 

61470 

43100 

70530 

110830 

20000 

70.1 

3-321 

2.029 

I.2QI 

7.216 

59180 

4*45° 

67840 

106620 

19080 

70.0 

3-534 

2.232 

1.302 

7.216 

59180 

41820 

66210 

113500 

20480 

70-7 

3-534 

2.232 

1.302 

7.216 

59l8o 

42760 

67710 

116070 

20940 

72-3 

3-645 

2.369 

1.276 

7  216 

58170 

44650 

68700 

127550 

22550 

76.8 

3-675  . 
5.125 

2.389 
2.084 

1.286 
I.O4T 

7.216 
6.013 

64170 
59*80 

4305o 
41310 

66230 
61960 

123030 
124020 

21930 
21470 

67.1 
69.8 

3  3°9 

2.206 

I.I03 

6.013 

61470 

42000 

62990 

125990 

23IIO 

68.3 

2.8b6 

1.428 

1.428 

9-425 

62660 

40620 

81230 

81230 

I23IO 

64.8 

2.976 

1.488 

1.488 

9-425 

61740 

36290 

72580 

72580 

11460 

58.8 

3.060 

i  .620 

1.440 

9-425 

62660 

38660 

73020 

82150 

"550 

61.7 

3.088 

1.636 

1.452 

9-425 

62660 

38000 

71730 

80820 

12450 

60.6 

3.378 

1.878 

1.500 

9-425 

61470 

37800 

68000 

85130 

13550 

61.5 

3-375 

1-875 

1.500 

9-425 

61470 

39000 

70200 

87750 

13970 

63-4 

604 


APPLIED    MECHANICS. 


TABULATION   OF   SINGLE- 
i"  STEEL 


No. 
of 
Test. 

Sheet  Letters. 

Pitch. 

No. 
of 
Rivets. 

Width 
of 
Joint. 

Nominal  Thick- 
ness. 

Size  of 
Rivets 
and 
Holes. 

Actual 
Thick- 
ness of 
Plate. 

Lap. 

Plate. 

Covers. 

Plate. 

Covers. 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

1344 

L 

D 

E 

*t 

6 

14.24 

1/4 

3/i  6 

lf*i 

.250 

2 

J345 

H 

D 

E 

14 

6 

" 

" 

44 

44 

.247 

2 

1346 

I 

E 

E 

2* 

6 

15.00 

" 

44 

44 

.251 

2 

1347 

I 

E 

E 

" 

6 

" 

" 

" 

*• 

.252 

2 

1348 

H 

c 

C 

2f 

5 

13-13 

" 

it 

it 

.244 

2 

J349 

G 

C 

C 

44 

5 

" 

" 

44 

*• 

•239 

2 

I35<> 

N 

D 

D 

a* 

5 

13-77 

" 

it 

M 

.250 

2 

I35i 

N 

D 

D 

44 

5 

" 

" 

44 

44 

.249 

2 

1352 

H 

D 

D 

2* 

5 

14-39 

" 

44 

II 

•  247 

2 

1353 

H 

E 

E 

ii 

5 

(i 

ii 

ii 

|| 

.248 

2 

T354 

I 

E 

E 

3 

5     . 

15.00 

(i 

14 

II 

.252 

2 

J355* 

I 

E 

E 

" 

5 

" 

44 

44 

(I 

•251 

2 

STEEL 


I356 

A 

I 

I 

if 

6 

9-75 

3/8 

i/4 

A*i 

-365 

2 

1357 

A 

I 

I 

44 

6 

44 

44 

44 

44 

.364 

2 

1358 

A 

I 

J 

i* 

6 

10.49 

K 

44 

44 

•  365 

2 

1359 

A 

J 

J 

44 

6 

" 

44 

44 

44 

.366 

2 

1360 

A 

I 

J 

44 

6 

10.50 

44 

11 

tt*l 

.366 

2 

1361 

A 

I 

I 

44 

6 

" 

41 

11 

44 

•  367 

2 

1362 

A 

J 

J 

i* 

6 

".25 

44 

44 

44 

.366 

2 

*3€>3 

A 

J 

J 

44 

6 

44 

(4 

41 

44 

•  365 

2 

1364 

B 

K 

K 

2 

6 

12.00 

II 

K 

44 

.388 

2 

1^65 

B 

K 

K 

II 

6 

44 

44 

44 

44 

•39° 

2 

1366 

C 

K 

K 

2* 

6 

12.76 

44 

44 

41 

.367 

2 

1367 

B 

K 

L 

44 

6 

44 

44 

11 

44 

•387 

2 

1368 

A 

J 

J 

it 

6 

11.25 

44 

44 

13*1 

-369 

2 

1369 

A 

J 

J 

44 

6 

14 

11 

It 

44 

.366 

2 

1370 

B 

K 

K 

2 

6 

12.  OO 

44 

It 

44 

-389 

2 

i37i 

B 

J 

J 

44 

6 

" 

44 

II 

41 

•388 

2 

i372 

B 

L 

L 

at 

6 

"-77 

44 

" 

44 

-385 

2 

1373 

C 

K 

L 

44 

6 

41 

It 

(I 

ii 

-367 

2 

1374 

D 

H 

N 

2i 

6 

13.5° 

44 

11 

44 

.376 

2 

1375 

E 

N 

N 

" 

6 

11 

K 

14 

44 

.380 

2 

1376 

D 

L 

•  M 

2| 

6 

14.23 

u 

11 

ii 

•383 

2 

1377 

H 

L 

M 

44 

6 

" 

" 

" 

it 

•37i 

2 

*  Fractured  two  outside  sections  of  plate  at  each  edge  along  line 


TABULATION  OF  RIVETED  JOINTS 


605 


RIVETED    BUTT-JOINTS— Continued. 


PLATE—  Continued. 


Sectional  Area 
of  Plate. 

Bearing 
Surface 
of 
Rivets. 

Shear- 
ing 
Area  of 
Rivets. 

Tensile 
Strength 
of  Plate 
per 
Sq.  In. 

Maximum  Stress  on  Joint  per  Sq.  In. 

Effi- 
ciency 
of 
Joint. 

.  Tension 
on  Gross 
Section  of 
Plate. 

Tension 
on  Net 
Section  of 
Plate. 

Comp.  on 
Bearing 
Surface 
of  Rivets. 

Shear- 
ing  of 
Rivets. 

Gross. 

Net. 

sq.  in. 

sq.  in. 

sq.  in. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

3-56o 

2.060 

1.500 

9-425 

61470 

3913° 

67640 

92870 

14780 

63.6 

3-520 

2.038 

1.482 

9-425 

59180 

-  40450 

69860 

96070 

15110 

68.3 

3-765 

2.259 

1.506 

9-425 

60480 

4359° 

72640 

108960 

17410 

72.1 

3.780 

2.268 

1.512 

9-425 

60480 

41420 

69030 

103540 

16610 

68.5 

3.204 

1.984 

i  .220 

7-854 

59180 

38700  - 

62490 

101620 

!5790 

65.4 

3-i3i 

1.936 

1-195 

7-854 

62660 

42890 

69370 

112380 

17100 

68.4 

3-442 

2.  192 

1.250 

7-854 

5374° 

42960 

67460 

118300 

18830 

7  j.i 

3.426 

2.181 

1-245 

7-854 

55740 

41780 

65640 

114980 

18230 

74-9 

3-554 

2-3!9 

1-235 

7-854 

59i8o 

43510 

66690 

125220 

19690 

73-5 

3-569 

2.329 

1.240 

7-854 

59i8o 

4383° 

67170 

126160 

19920 

74-1 

3.780 

2.520 

1.260 

7-854 

7    Rcj 

60480 
60480 

44580 

66870 

66610 

133730 

2I450  j      73.7 

3'  7^5 

7  •  °54 

444^0 

133230 

21290         73-4 

PLATE. 


3-559 

2.190 

1.369 

3.682 

54260 

40460 

65740 

105170 

39100 

74.6 

3-549 

2.184 

1-365 

3.682 

54260 

39420 

64060 

102490 

38000 

72.6 

3-829 

2.460 

1.369 

3.682 

54260 

39780 

61910 

111250 

41360 

73-3 

3-843 

2.471 

1.372 

3.682 

54260 

39060 

60740 

109400 

40770 

72.0 

3-843 

2.196 

1.647 

5-300 

54260 

37000 

64750 

86330 

26830 

68.2 

3-854 

2.  2O2 

1.652 

5-300 

54260 

37050 

64840 

86430 

26940 

68.3 

4.118 

2.471 

1.647 

5.30° 

54260 

37450 

62400 

93620 

29090 

69.0 

4.106 

2.464 

1.642 

5-300 

54260 

38040 

63390 

95130 

.29470 

70.1 

4-656 

2.910 

1.746 

5-3oo 

59730 

41820 

66910 

111510 

3673° 

70.0 

4.680 

2.925 

1-755 

5-3oo 

59730 

42000 

67200 

II2OOO 

37090 

70-3 

4.683 

3-031 

1.652 

5.300 

57870 

41040 

63410 

116340 

36260 

70.9 

4-938 

3-197 

1.741 

5-3°° 

59730 

40910 

63180 

116030 

38110 

68.5 

4-i5i 

2.214 

J-937 

7.216 

54260 

35000 

65620 

75000 

20130 

64-5 

4.114 

2.192 

1.922 

7.2,6 

54260 

34180 

64140 

73150 

19480 

63.0 

4.668 

2.626 

2.042 

7  216 

59730 

36870 

65540 

84280 

23850 

61.7 

4.656 

2    619 

2.037 

7.216 

59730 

38940 

69220 

SgOOO 

25160 

65-2 

4.916 

2.895 

2.021 

7.216 

59730 

38730 

65770 

94210 

26390 

64.8 

4.672 

2-745 

1.927 

7.216 

57870 

39010 

65660 

94580 

25260 

67.4 

5.076 

3.102 

1-974 

7.216 

53730 

37960 

62120 

97620 

26700 

70.6 

5-130 

3-^35 

1-995 

7.216 

5834° 

39810 

65140 

102360 

28300 

68.2 

5-450 

3-439 

2.  Oil 

7.216 

53730 

38920 

61670 

105470 

29390 

72-4 

5.290 

3-343 

1.948 

7.216 

56670 

39870 

6311O 

108260 

29230 

70.4 

of  riveting  ;  the  two  middle  sections  sheared  in  front  ^f  rivets. 


6o6 


APPLIED    MECHANICS. 


TABULATION    OF    SINGLE- 
S'' STEEL 


Sheet  Letters. 

Nominal 

Thickness. 

No.  of 
Test. 

Pitch. 

No.  of 
Riv- 
ets. 

Width 
of 
Joint. 

Size  of 
Rivets 
and 
Holes. 

Actual 
Thick- 
ness of 
Plate. 

Lap. 

Plate. 

Covers. 

Plate. 

Covers. 

1378 

D 

M 

M 

in. 

2* 

.6 

in. 
15.00 

in. 

3/8 

in. 
i/4 

in. 

li** 

in. 

•383 

in. 

2 

1379*  1     D 

M 

M 

" 

6 

" 

it 

" 

" 

.385 

2 

1380 

B 

J 

K 

2 

6 

12.00 

" 

" 

if*  ' 

•  388 

2 

1381 

E 

K 

K 

" 

6 

" 

" 

" 

" 

.381 

2 

1382 

B 

K 

K 

2* 

6 

12.75 

" 

" 

" 

.388 

2 

i383t 

E 

G 

K 

" 

6 

" 

11 

11 

" 

•383 

2 

1384 

F 

H 

H 

2± 

6 

13-49 

M 

it 

" 

.381 

2 

1385 

E 

N 

N 

" 

6 

" 

" 

(i 

*« 

.380 

2 

1386* 

H 

L 

L 

2f 

6 

14.25 

" 

it 

" 

.368 

2 

1387 

G 

L 

M 

" 

6 

tt 

" 

" 

" 

•365 

2 

1388 

D 

M 

M 

** 

6 

15.00 

11 

" 

" 

•385 

2 

1389 

D 

M 

M 

" 

6 

" 

" 

" 

" 

.386 

2 

1390 

C 

G 

G 

2| 

5 

13.12 

" 

" 

" 

•372 

2 

MQI§ 

C 

H 

L 

" 

5 

" 

" 

" 

M 

•369 

2 

1302 

C 

L 

L 

2j 

5 

13-75 

" 

«« 

" 

•374 

2 

1393 

C 

L 

N 

" 

5 

" 

K 

" 

" 

•372 

2 

J394§ 

D 

I 

M 

2| 

5 

M-39 

tl 

it 

" 

.386 

2 

1395 

D 

M 

M 

" 

5 

"   i     " 

1C 

" 

•383 

2 

*  Test  discontinued  soon  after  passing  maximum  load. 
+  Test  discontinued  at  maximum  load. 
$  Test  discontinued  after  passing  maximum  load. 
§  Test  discontinued  before  fracture  was  complete. 


TABULATION    OF  RIVETED  JOINTS. 


607 


RIVETED    BUTT-JOINTS— Continued. 


PLATE—  Continued. 


Sectional  Area 

Maximum  Stress  on  Joint  per 

of  Plate. 

Square  Inch. 

Tensile 

Bearing 

Shear- 

Strength 

Effi- 

Surface 
of 

ing 
Area  of 

of  Plate 
per 

Tension 

Compres- 
Tension      sion  on 

ciency 
of 

Plate. 

Covers. 

Rivets. 

Rivets. 

Square 
Inch. 

on  Gross 
Section 

on  Net 
Section 

Bearing 
Surface 

Shear- 
ing of 

Joint. 

oi  Plate. 

of  Plate. 

of 

Rivets. 

Rivets. 

sq.  in. 

sq.  in. 

sq.  in. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

tos. 

Ibs. 

5  745 

3-734 

2.  Oil 

7.216 

53730 

40560 

62400 

115860 

32290 

75-5 

5  775 

3-754 

2.021 

7.216 

5373° 

40700 

62620 

116280 

3257o 

75-7 

4656 

2.328 

2.328 

9-425 

5973° 

345°° 

69010 

69010 

17050 

57-8 

4-572 

2.286 

2.286 

9.425 

58340 

33440 

66880 

66880 

16220 

57-3 

4-947 

2.619 

2.328 

9-425 

59730 

35590 

67230 

75640 

18680 

59-6 

4.883 

2-585 

2.298 

9-425 

58340 

34730 

65610 

73800 

17990 

59-5 

5.140 

2.854 

2.286 

9-425 

54290 

3549° 

63930 

79810 

19360 

65.4 

5126 

2.846 

2.280 

9-425 

58340 

35840 

64550    i      80570 

19490 

61.4 

5-244 

3.036 

2.208 

9-425 

56670 

37010 

63930 

87910 

20590 

65-3 

5-205 

3-oi5 

2.190 

9.425 

53840 

36750 

63450 

87350 

20300 

68.2 

5-775 

3-465 

2.310 

9-425 

53730 

3749° 

62480 

937io 

22970 

69.8 

5-79° 

3-474 

2.3l6 

9-425 

53730 

3736o 

62260 

93390 

21890 

69.5 

4.881 

3-021 

1.  860 

7.854 

57870 

39000 

63010 

102340 

24240 

67-4 

4.841 

2.996 

1.845 

7-854 

57870 

39520 

63850 

103690 

24360 

68.3 

5-143 

3-273 

I.SjO 

7-854 

57870 

39840 

62590 

109540 

26080 

68.9 

5.111 

3251 

1.860 

7-854 

57870 

40420 

63550 

111070 

26310 

69.8 

5o55 

3-625 

1.930 

7-854 

53730 

3934° 

60290 

113240 

37830 

73-2 

5-496 

3.58i 

I-9IS 

7-854 

5373° 

40300 

eiseo 

115660 

28200 

75-o 

6O8  APPLIED  MECHANICS. 


SINGLE-RIVETED    BUTT-JOINTS,  STEEL   PLATE. 
DESCRIPTION    OF    TESTS    AND    DISCUSSION    OF    RESULTS. 

"  The  following  tests  complete  a  series  of  two  hundred  and 
sixteen  single-riveted  butt-joints  in  steel  plates,  in  which  the 
thickness  of  the  plates  ranged  from  J"  to  f " ,  and  the  size  of 
the  rivets  from  -fa"  to  ly3^"  diameter. 

The  plates  were  annealed  after  shearing  to  size,  the  edges 
opposite  the  joint  milled  to  the  finished  width  ;  the  holes  were 
drilled  and  rivets  machine-driven.  Iron  rivets  were  used 
throughout,  except  in  some  of  the  fx/  joints. 

Tensile  tests  of  the  plates  and  rivet-metal,  together  with 
the  tests  of  the  joints  in  £"  and  f "  plate,  are  contained  in  the 
Report  of  Tests  of  1885,  Senate  Document  No.  36,  Forty-ninth 
Congress,  first  session. 

The  tests  herewith  presented  comprise  the  details  and  tab- 
ulation of  joints  in  £",  f",  and  f-"  thickness  of  plate,  a  portion 
of  which  were  tested  hot. 

The  gauged  length  in  which  elongations  and  sets  were 
measured  was  5";  2\"  each  side  of  the  centre  line  of  the  joint. 

During  the  progress  of  testing  the  same  characteristics  were 
displayed  which  were  referred  to  in  the  previous  report.  The 
joints  were  very  rigid  under  the  early  loads.  This  rigidity  is 
overcome  by  loads  which  exceed  the  friction  between  the  plate 
and  covers,  after  which  the  stretching  proceeded  slowly  with 
some  fluctuations  till  elongation  of  the  metal  of  the  net  section 
became  general  ;  the  metal  under  compression  in  front  of  the 
rivets  yielding,  also  the  rivets  themselves. 

The  behavior  of  joints  in  different  thicknesses  of  plate  is 
substantially  the  same,  and  an  examination  of  the  results  shows 
that  when  exposed  to  similar  conditions  the  strength  per  unit 


SIXGLE-RIVETED  BUTT-JOINTS,    STEEL   PLATE.      '609 

of  fractured  metal  is  nearly  the  same,  whether  J"  or  J"  plate  is 
used. 

It  will  not  be  understood  from  this,  however,  that  as  a  con- 
sequence the  same  efficiency  may  be  obtained  in  different 
thicknesses  of  plate  for  single-riveted  work,  because  it  will  be 
seen  that  certain  essential  conditions  change  as  we  approach 
the  stronger  joints  in  different  thicknesses  of  plate. 

A  riveted  joint  of  the  maximum  efficiency  should  fracture 
the  plate  along  the  line  of  riveting,  for  it  is  clear  that  if  failure 
occurs  in  any  other  manner,  as  by  shearing  the  rivets  or  tear- 
ing out  the  plate  in  front  of  the  rivet-holes,  there  remains  an 
excess  of  strength  along  the  line  of  riveting,  or  in  other  words 
along  the  net  section  of  metal — if  in  a  single-riveted  joint — • 
which  has  not  been  made  use  of ;  but  when  fracture  occurs, 
along  the  net  section  an  excess  of  strength  in  other  directions 
is  immaterial. 

If  the  strength  per  unit  of  metal  of  the  net  section  was  con. 
stant,  it  would  be  a  very  simple  matter  to  compute  the  effi- 
ciency of  any  joint,  as  it  would  merely  be  the  ratio  of  the  net 
to  the  gross  areas  of  the  plates. 

The  tenacity  of  the  net  section,  however,  varies,  and  this 
variation  extends  over  wide  limits. 

In  the  present  series  there  is  an  excess  in  strength  of  the 
net  section  over  the  strength  of  the  tensile  test-pieces  in  all 
joints. 

Special  tables  have  been  prepared  showing  this  behavior. 

The  efficiencies  shown  in  Table  No.  I  are  obtained  by  divid- 
ing the  tensile  stress  on  the  gross  area  of  plate  by  the  tensile 
strength  of  the  plate  as  represented  by  the  strength  of  the  ten- 
sile test-strip,  stating  the  values  in  per  cent;  of  the  latter. 

Table  No.  2  exhibits  the  differences  between  the  efficien- 
cies of  the  joints  and  the  ratios  of  net  to  gross  areas  of  plate. 
If  the  tenacity  of  net  section  remained  constant  per  unit  of 


APPLIED  MECHANICS. 


area,  the  efficiencies  in  Table  No.  I  would,  as  above  explained, 
be  identical  with  the  ratios  of  net  to  gross  areas  of  plate,  and 
the  values  in  this  table  reduced  to  zero. 

Table  No.  3  shows  the  excess  in  strength  of  the  net  section 
of  the  joint  over  the  strength  of  the  tensile  test-strip  in  per 
cent  of  the  latter. 

Table  No.  4  exhibits  the  compression  on  the  bearing-surface 
of  the  rivets  in  connection  with  the  excess  in  tensile  strength 
of  the  net  section  of  plate. 

Table  No.  I  is  valuable  in  showing  at  once  the  value  of 
different  joints  wherein  the  pitch  of  the  rivets  and  their  diame- 
ters vary. 

It  is  seen  there  is  considerable  latitude  allowed  in  the  choice 
of  rivets  and  pitch  without  materially  changing  the  efficiency 
of  the  joint ;  thus  in  J"  plate, 

f"  rivets  (driven),  if"  pitch,  72.4  per  cent  efficiency, 

f"  rivets  (driven),  i\"  pitch,  73.3  per  cent  efficiency, 

f-"  rivets  (driven),  af"  pitch,  71.5  per  cent  efficiency, 

i"  rivets  (driven),  2-3-"  pitch,  70.3  per  cent  efficiency, 

i"  rivets  (driven),  2^"  pitch,  73.8  per  cent  efficiency, 

give  nearly  the  same  results. 

In  these  examples  the  ratios  of  net  to  gross  areas  of  plate 
range  from  60  to  67  per  cent,  while  the  rivet-areas  range  from 
.3067  square  inch  to  .7854  square  inch.  The  actual  areas  of 
net  sections  of  plate  and  rivets  are  as  follows : 


|"  rivets. 

\"  rivets. 

\"  rivets. 

i"  rivet*. 

Rivets. 

sq.  in. 
.  ^067 

sq.  in. 
.4418 

sq.  in. 
.6013 

sq.  in. 

.7854 

Plate  .     . 

1.486 

2.  2O7 

2.2^2 

j  2.259 

(  2.319 

SINGLE-RIVETED  BUTT-JOINTS,    STEEL   PLATE.       6l  I 


The  areas  of  the  rivets  stand  to  each  other  as  the  following 
numbers  : 


100 


144 


.96 


and  the  net  areas  of  the  plate  to  each  other  as 
100  149  150 


256 


( 152 
1 156 


From  these  illustrations  it  appears  that  to  attain  the  same 
degree  of  efficiency  in  this  quality  of  metal,  although  that 
efficiency  is  probably  not  the  highest  attainable,  a  fixed  ratio 
between  rivet  metal  and  net  section  of  plate  is  not  essential. 

In  \"  plate  with  \"  rivets  the  efficiencies  of  the  joints  tested 
cold  are  nearly  constant  over  the  range  of  pitches  tested. 

The  efficiencies  and  the  ratio  of  net  to  gross  areas  of  plate 
are  as  follows  : 


Pitch. 

4" 

•2" 

-4" 

2i" 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

Efficiency.     . 
Ratio  of  areas  .     . 

64.5 
53-4 

66.3 
56.3 

66.3 
58.9 

66.4 
6l.I 

In  this  we  have  illustrated  a  case  which,  in  passing  from 
the  widest  pitch,  having  61.1  per  cent  of  the  solid  plate  left,  to 
the  narrowest  pitch,  which  had  53.4  per  cent  of  the  solid  plate, 
the  gain  or  excess  in  strength  in  the  net  section  almost  exactly 
compensated  for  the  loss  of  metal. 

In  Table  No.  3  the  average  of  all  the  joints  shows  the  high- 
est per  cent  of  excess  of  strength  in  the  narrowest  pitch,  and  a 
tendency  to  lose  this  excess  as  the  pitch  increases. 

Tests  of  detached  grooved  specimens  show  the  same  kind 


612 


APPLIED   MECHANICS. 


of  behavior,  but  as  they  are  not  subject  to  all  the  conditions 
found  in  a  joint,  the  analogy  does  not  extend  very  far. 

The  maximum  gain  in  strength  on  the  net  section,  not  for 
the  time  being  regarding  the  hot  joints,  and  disregarding  the 
•exceptionally  high  value  of  joint  No.  1339,  J-"  plate,  was  21.2 
per  cent,  the  minimum  value  2.5  per  cent  of  the  tensile  test- 
:strip.  In  other  forms  of  joints,  and  with  punched  holes  in  both 
iron  and  steel  plate,  illustrations  are  numerous  in  which  there 
have  been  large  deficiencies,  the  metal  of  the  net  section  fall- 
ing far  below  the  strength  of  the  plate. 

It  is  believed  to  have  been  amply  shown  that  increasing 
the  net  width  diminishes  the  apparent  tenacity  of  the  plate, 
although  other  influences  may  tend  to  counteract  this  tendency 
in  some  joints. 

In  order  to  compare  the  excess  in  strength  of  one  thickness 
of  plate  with  another  having  the  same  net  widths,  we  have  the 
following  table,  rejecting  those  joints  that  failed  otherwise 
than  along  the  line  of  riveting  in  making  these  averages:  — 


Thickness  of  Plate. 

Width  of  plate  between  rivet-holes. 

i" 

Ii" 

i}" 

if" 

I*" 

If" 

If" 

i*" 

*    2" 

y  

i-' 

P.  ct. 

16.7 
18.4 
16.7 
17.7 
11.4 

P.  ct. 

12.6 

13-7 
14-3 

16.3 

JS-i 

P.  ct. 

11.4 
12.7 

9-3 
14.2 
13.8 

P.  ct. 

12.0 
13-5 
10-7 
14-5 
I4.I 

P.  Ct. 

13-4 
I4.6 
Q.I 
I4.6 
7.6 

P.  Ct. 

8.9 

12.9 

8.8 
12.7 

IT.  8 

P.  ct. 
"•5 

9.0 

8.2 

9-9 

IO.O 

P.   Ct. 

13-1 

13.6 

12.2 

0.8 
10.  i 

ii.  8 

P.  Ct. 

10.6 

3-5 

£'        

i'     

!/  

Average    of   all    thick- 
nesses     

16.2 

14.4 

12.3 

12.  Q 

II.9 

II.  0 

9-7 

7.0 

The  excess  in  strength  is  generally  well  maintained  in  each 
of  the  several  thicknesses,  and  were  it  possible  to  retain  the 
same  ratio  of  net  to  gross  areas  of  plate,  and  at  the  same  time 


SINGLE-RIVETED  BUTT-JOINTS,    STEEL  PLATE.       613 


equal  net  widths  between  rivets,  it  would  seem  from  this  point 
of  view  feasible  to  obtain  the  same  degree  of  efficiency  in  thick 
as  in  thin  plates. 

The  following  causes,  however,  tend  to  prevent  such  a  con 
summation. 

For  equal  net  widths  thick  plates  require  larger  rivets  to 
avoid  shearing  than  thin  ones,  the  diameters  of  the  rivets  being 
somewhat  increased  for  this  cause,  and  again  because  it  has 
become  necessary  to  increase  the  metal  of  the  net  section  in 
order  to  retain  a  suitable  ratio  of  net  to  gross  areas  of  plate. 

There  results  from  these  considerations  such  an  increase  in 
net  width  of  plate  that  the  excess  in  strength  displayed  by 
narrower  sections  is  lost,  and  consequently  the  result  is  a  joint 
of  lower  efficiency. 

The  data  relating  to  the  influence  of  compression  on  the 
bearing-surface  of  the  rivets,  on  the  tensile  strength  of  the 
plate,  as  shown  by  Table  No.  4  are  more  or  less  conflicting. 
However,  in  the  J"  plate,  in  which  the  most  intense  pressures 
are  found,  there  is  seen  a  pronounced  increase  in  tensile  strength 
as  the  pressures  diminish  in  intensity. 

It  is  probable  that  the  effects  of  intense  compression  would 
be  more  conspicuous  in  a  less  ductile  metal,  or  one  in  which 
the  ductility  had  been  impaired  by  punched  holes  or  otherwise. 

A  number  of  joints  were  tested  at  temperatures  ranging 
between  200°  and  703°  Fahr. 

The  heating  was  done  after  the  joints  were  in  position  for 
testing,  by  means  of  Bunsen  burners,  arranged  in  a  row  par- 
allel to  and  under  the  line  of  riveting. 

The  temperature  was  determined  with  a  mercurial  ther- 
mometer, the  bulb  of  which  was  immersed  in  a  bath  of  oil, 
contained  in  a  pocket  drilled  in  the  middle  rivet  of  the  joint. 

When  at  the  required  temperature  the  thermometer  was 
removed  from  the  joint,  a  dowel  was  driven  into  the  pocket  to 


6  14  APPLIED  MECHANICS. 

compensate  for  the  metal  of  the  rivet  which  had  been  removed 
by  the  drill,  and  then  loads  applied  and  gradually  increased  up 
to  the  time  of  rupture. 

Three  joints,  Nos.  1423,  1426,  and  1430,  were  tested  with- 
out dowels  in  the  oil-pockets. 

The  method  of  heating  was  to  raise  the  temperature  of  the 
joint,  as  shown  by  the  thermometer,  a  few  degrees  above  the 
temperature  at  which  the  test  was  made,  shut  off  the  gas- 
burners,  and  allow  the  temperature  to  fall  to  the  required  limit. 
The  temperature  fell  slowly,  draughts  of  cold  air  being  excluded 
from  the  under  side  of  the  joint  by  the  hood  which  covered 
the  gas-burners ;  the  upper  side  and  edges  of  the  joint  were 
covered  with  fine  dry  coal-ashes. 

The  results  show  an  increase  in  tensile  strength  when  heated 
over  the  duplicate  cold  joints  at  each  temperature  except  200° 
Fahr. 

From  200°  there  was  a  gain  in  strength  up  to  300°,  when 
the  resistance  fell  off  some  at  350°,  increased  again  at  400°,  and 
reached  the  maximum  effect  observed  at  500°  Fahr. ;  from  this 
point  the  strength  fell  rapidly  at  600°  and  700°. 

In  per  cent  of  the  cold  joint  there  was  a  loss  at  200°  of  3.2 
per  cent,  the  average  of  three  joints  ;  at  500°  the  gain  was  22.6 
per  cent,  the  average  of  four  joints.  The  maximum  and  mini- 
mum joints  at  this  temperature  showed  gains  of  27.6  per  cent, 
and  18.3  per  cent,  respectively. 

The  highest  tensile  strength  on  the  net  section  of  plate  was 
found  in  joint  No.  1433,  tested  at  500°  Fahr.,  where  81050 
pounds  per  square  inch  was  reached  against  a  strength  of  58000 
pounds  per  square  inch  in  the  cold  tensile  test-strip. 

The  hot  joints  showed  less  ductility  than  the  cold  ones, 
those  tested  at  200°  Fahr.  not  being  exempt  from  this  behav- 
ior, although  there  was  no  near  approach  to  brittleness  in  any. 

Three   joints,    Nos.    1418,   1420,  and    1424,   were   heated; 


SINGLE-RIVETED  BUTT-JOINTS,    STEEL   PLATE.       615 

strained  when  hot  with  loads  exceeding  the  ultimate  strength 
of  their  duplicate  cold  joints;  the  loads  were  released,  and 
after  having  cooled  to  the  temperature  of  the  testing-room 
(No.  1424  cooled  to  150°  Fahr.)  were  tested  to  rupture,  and 
were  found  to  have  retained  substantially  the  strength  due 
their  temperature  when  hot. 

In  order  to  ascertain  that  the  time  intervening  between  hot 
straining  and  final  rupture  did  not  contribute  towards  the  ele- 
vation in  strength,  joint  No.  1434  was  strained  in  a  similar 
manner  with  a  load  approaching  rupture,  after  which  a  period 
of  rest  was  allowed  and  then  ruptured  without  material  gain 
in  strength. 

A  peculiarity  of  the  joints  fractured  at  400°  and  higher  tem- 
peratures was  the  comparatively  smooth  surface  of  the  frac- 
tured sections,  and  which  took  place  in  planes  making  angles 
of  about  50°  with  the  rolled  surface  of  the  plate. 

The  shearing-strength  of  the  iron  rivets  was  also  increased 
by  an  elevation  of  temperature. 

The  rivets  in  joint  No.  1410  at  the  temperature  of  350° 
sheared  at  43060  pounds  per  square  inch,  while  in  the  dupli- 
cate cold  joint  No.  1411  they  sheared  at  38530  pounds  per 
square  inch,  and  the  rivets  in  pint  No.  1398  at  300°  Fahr. 
were  loaded  with  46820  pounds  per  square  inch  and  did  not 
shear. 

Other  examples,  where  some  of  the  rivets  sheared  and  the 
plate  fractured  in  part,  showed  corresponding  gains  in  shearing- 
strength. 

The  almost  entire  absence  of  granular  fractures  in  these 
tests  is  a  feature  too  important  to  pass  by  without  special  men 
tion." 


6i6 


APPLIED    MECHANICS. 


TABULATION   OF   SINGLE- 
STEEL   PLATE. 


No. 
of 
Test. 

Sheet  Letters. 

Pitch. 

No. 
of 
Rivets. 

Width 
of 
Joint. 

Nominal  Thick- 
ness. 

Size  of 
Rivets 
and 
Holes. 

Actual 
Thick- 
ness of 
Plate. 

Lap. 

Plate. 

Covers. 

Plate. 

Covers. 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

1396 

R 

R 

R 

'* 

6 

10.50 

1/2 

5/i6 

tt** 

.481 

2 

J397 

R 

R 

R 

" 

6 

" 

" 

" 

" 

.484 

2 

1398 

R 

R 

R 

«t 

6 

11.25 

" 

" 

" 

484 

2 

1399 

R 

R 

R 

" 

6 

" 

11 

it 

11 

-483 

2 

1400 

R 

R 

R 

2 

6 

13.00 

•• 

" 

ti 

.486 

2 

1401 

R 

S 

T 

" 

6 

" 

" 

" 

44 

-483 

2 

1402 

R 

R 

R 

4 

6 

11.25 

" 

" 

ii*l 

.481 

2 

1403 

R 

R 

R 

" 

6 

" 

" 

11 

" 

.486 

2 

1404 

R 

R 

R 

2 

6 

12.  OO 

•' 

" 

" 

.486 

2 

1405 

R 

S 

S 

" 

6 

" 

" 

44 

" 

•  487 

2 

1406 

S 

S 

S 

a* 

6 

12.75 

" 

M 

11 

.470 

2 

1407 

S 

S 

S 

" 

6 

" 

n 

" 

ti 

.471 

2 

1408 

T 

Q 

Q 

at 

6 

I3-SO 

" 

it 

it 

.486 

2 

1409 

T 

Q 

Q 

'• 

6 

" 

11 

•• 

" 

.482 

2 

1410 

T 

0 

0 

af 

6 

14.25 

44 

it 

" 

.481 

2 

1411 

T 

0 

o 

" 

6 

It 

it 

tt 

" 

.485 

2 

1412 

R 

R 

R 

2 

6 

12.00 

" 

" 

IS*  i 

.484 

2 

1413 

R 

R 

R 

" 

6 

" 

it 

" 

11 

.481 

2 

14.4 

S 

S 

S 

at 

6 

12-75 

44 

it 

11 

•472 

2 

«4'5 

S 

S 

S 

" 

6 

11 

*' 

11 

" 

.468 

2 

,416 

S 

Q 

Q 

at 

6 

JS-SO 

" 

" 

" 

.468 

2 

1417 

T 

Q 

Q 

" 

6 

u 

44 

it 

it 

.482 

2 

1418 

T 

0 

0 

a| 

6 

14-25 

" 

" 

it 

.481 

2 

14-9 

T 

o 

0 

" 

6 

" 

11 

ti 

11 

.482 

2 

1420 

U 

p 

p 

2* 

6 

15.00 

11 

" 

it 

•479 

2 

1421 

u 

p 

p 

" 

6 

" 

" 

M 

" 

.483 

2 

1422 

S 

S 

S 

a* 

5 

I3-I3 

" 

it 

11 

.469 

2 

1423 

S 

S 

S 

" 

5 

" 

l< 

" 

it 

•473 

2 

1424 

T 

0 

0 

a* 

5 

13-75 

" 

44 

•' 

.484 

2 

1425 

T 

o 

o 

" 

5 

" 

" 

" 

" 

-483 

2 

1426 

S 

S 

S 

at 

6 

12.75 

11 

" 

xA*ii 

•474 

2 

1427 

R 

S 

S 

** 

6 

" 

" 

ct 

44 

•475 

2 

1428 

T 

Q 

0 

at 

6 

I3-50 

" 

" 

11 

•479 

2 

1429 

S 

0 

Q 

" 

6 

" 

" 

14 

" 

•465 

2 

'43° 

U 

o 

o 

«f 

6 

14-25 

" 

44 

44 

-484 

2 

M3i 

U 

0 

o 

" 

6 

" 

11 

" 

" 

•483 

2 

I 

TABULATION   OF  RIVETED  JOINTS. 


6l7 


RIVETED    BUTT-JOINTS. 


STEEL   PLATE. 


Sectional  Area 
of  Plate. 

Bearing 

Surface 
nf 

Shear- 
ing 

Tensile 
Strength 
of  Plate 

Max.  Stress  on  Joint  per  Sq.  In. 

J| 

|.E| 

c  ^  ^o 

!-s!s 

§^£ 

•I'ga 

Gross. 

Net. 

OI 

Rivets. 

Area  of 
Rivets. 

per 
Sq.  In. 

|c|l 

^  c  o  5"! 

C-N   O   Q> 

all  5 

J3  £££ 

1? 

T*'c 

°c/) 

C/J 

O             "Q 

1—1 

sq  in. 

sq.  in. 

sq.  in. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

5-051 

2.886 

2.165 

5-301 

57180 

37750 

66070 

88080 

35970 

66.1 

2OO 

5.082 

2.904 

2.178 

5-3°* 

57180 

38980 

68220 

90960 

37370 

68.1 

5-445 

3.267 

2.178 

5-301 

57180 

45560. 

75960 

113960 

46820 

79-6 

300 

5-439 

3.266 

2-173 

5-3o: 

57180 

39260 

65400 

98290 

40290 

68.6 

5.842 

3-655 

2.187 

5-301 

57180 

37000 

59*40 

98830 

40770 

64-7 

S-796 

3.622 

2.174 

5-301 

57180 

39420 

63080 

105100 

43100 

68  9 

35° 

5-416 

2.891 

2-525 

7.216 

57180 

36890 

69110 

79*3° 

27690 

64-5 

5-477 

2.926 

2-551 

7.216 

57180 

38250 

71600 

81730 

29030 

66.9 

250 

5-832 

3.281 

2-55* 

7.216 

57180 

379*0 

67380 

86670 

30640 

66.3 

5-854 

3-297 

2-557 

7.216 

57180 

43730 

77650 

IOOI2O 

3548o 

76-4 

300 

5-997 

3-529 

2.468 

7.216 

59050 

44790 

76110 

108830 

37220 

76.0 

400 

6.010 

3-537 

2-473 

7.216 

59050 

39210 

66630 

953*0 

32660 

66.3 

6.561 

4.010 

2-551 

7.216 

60000 

39850 

65210 

102500 

36230 

66.4 

6.512 

3.982 

2-530 

7.216 

60000 

46610 

76220 

i  19980 

42060 

77  .6 

500 

6.859 

4  334 

2-525 

7.216 

60000 

45300 

71690 

123050 

43060 

75-5 

350 

6.916 

4-37° 

2.546 

7.216 

60000 

40050 

63390 

108800 

38530 

66.7 

5-813 

2.909 

2.904 

9-425 

57*80 

35920 

71770 

71900 

22150 

62.8 

250 

5-772 

2.886 

2.886 

9-425 

57*8o 

34390 

68780 

68780 

21060 

60.  i 

6.023 

3-i9i 

2.832 

9-425 

59050 

35000 

66020 

7443° 

22360 

59-2 

5-967 

3-159 

2.808 

9-425 

59050 

40250 

76030 

85540 

25480 

68.1 

300 

6.327 

3-5T9 

2.808 

9-425 

59050 

34660 

62320 

78090 

23160 

60.3 

200 

6.512 

3  .  620 

2.892 

9-425 

60000 

36950 

66480 

83220 

25530 

61.5 

6.859 

3-973 

2.886 

9-425 

60000 

437*o 

75460 

103880 

31810 

72.8 

(*) 

6.883 

3-991 

2.892 

9-425 

60000 

38720 

66770 

92*50 

28270 

64-5 

7-194 

4-320 

2-874 

9-425 

58000 

44840 

74670 

112250 

34230 

77-3 

(t) 

7-245 

4-347 

2.898 

9-425 

58000 

38740 

64570 

96850 

297Po 

66.9 

6.  .67 

3  822 

2-345 

7-854 

59050 

39730 

64110 

104490 

3T200 

67.. 

6  220 

3-855 

2-365 

7-854 

59050 

45420 

73250 

119450 

35840 

76.9 

400 

6.660 

4.240 

2.420 

7-854 

60000 

48950 

76890 

137110 

41510 

81.5 

(*) 

6  632 

4.217 

2-4*5 

7-854 

60000 

40600 

63860 

111490 

34280 

67.6 

6-053 

2-853 

3.200 

11.928 

59050 

35070 

74410 

66340 

18630 

59-3 

300 

6.042 

2.836 

3.206 

11.928 

59050 

30420 

64810 

5733° 

15410 

51.5 

6.471 

3-238 

3-233 

11.928 

60000 

40330 

80620 

80730 

21880 

67-2 

350 

6.278 

3-*39 

3-139 

11.928 

59°5° 

33420 

66840 

66840 

*759° 

56-5 

6.897 

3.620 

3-277 

11.928 

58000 

36390 

65150 

76590 

21040 

62.7 

700 

6.852 

3.632 

3.260 

11.928 

58000 

3366o 

63870 

71160 

1945° 

58.0 

*  Strained  while  at  temperature  of  400°  Fahr 
i  Strained  while  at  temperature  of  500°  Fahr. 

*  Strained  while  at  temperature  of  500°  Fahr 


,,  and  allowed  to  cool  before  rupture. 
,  and  allowed  to  cool  before  rupture. 
,,  then  cooled  to  150°  Fahr.,  and  ruptured. 


6i8 


APPLIED    MECHANICS. 


TABULATION   OF   SINGLE- 
STEEL  PLATE-  Continued. 


Sheet 

Nominal 

Letters. 

Thickness. 

No. 
of 
Test. 

5itch. 

No. 
of 
Riv- 
ets. 

Width 
of 
Joint. 

Size  of 
Rivets 
and 
Holes. 

Actual 
Thick- 
ness of 
Plate. 

Lap. 

Plate. 

Covers. 

Plate. 

Covers. 

1432 

u 

P 

P 

in. 

ai 

6 

in. 
15.00 

in. 

1/2 

in. 

5/i6 

in. 
iA*  il 

in. 

.484 

in. 

2 

M33 

U 

P 

P 

41 

6 

41 

44 

14 

44 

.481 

2 

M34 

R 

S 

Q 

2g 

5 

13-13 

44 

41 

44 

.472 

2 

*435 

S 

S 

S 

" 

5 

" 

44 

44 

44 

•475 

2 

1436 

T 

0 

o 

*! 

5 

13-75 

44 

44 

44 

.482 

2 

1437 

T 

0 

0 

11 

S 

" 

44 

" 

11 

.482 

2 

1438 

U 

P 

P 

aj 

5 

14-38 

it 

44 

f 

.484 

2 

1439 

U 

P 

P 

it 

5 

it 

44 

<t 

44 

.485 

2 

1440 

U 

P 

P 

3 

5 

15.00 

II 

44 

44 

.482 

2 

1441 

U 

P 

P 

it 

5 

44 

44 

" 

44 

•483 

2 

1442 

u 

P 

P 

34 

5 

15.68 

11 

41 

41 

•483 

2 

M43 

u 

P 

P 

" 

5 

" 

41 

" 

41 

.484 

2 

1444 

V 

E 

E 

I* 

6 

11.25 

5/8 

3/8 

11*5 

.621 

2 

1445 

V 

B 

E 

" 

6 

44 

it 

44 

41 

.624 

2 

1446 

V 

E 

E 

2 

6 

12.  OO 

44 

44 

44 

.616 

2 

M47 

V 

E 

E 

11 

6 

44 

44 

44 

** 

.624 

2 

1448 

V 

E 

E 

4 

6 

12  -75 

" 

44 

*' 

.621 

2 

M49 

V 

E 

E 

11 

6 

" 

44 

44 

44 

.624 

2 

145° 

w 

F 

G 

si 

6 

13-50 

44 

•4 

41 

.610 

2 

MS1 

w 

G 

G 

" 

6 

44 

44 

44 

44 

.611 

2 

1452 

V 

E 

E 

2 

6 

12.00 

44 

44 

if*  i 

.624 

2 

I4S3 

V 

E 

E 

'* 

6 

41 

" 

" 

** 

.620 

2 

M54 

V 

E 

E 

ai 

6 

"•75 

44 

44 

4< 

.622 

1 

1455 

V 

E 

E 

" 

6 

" 

tt 

44 

41 

.618 

2 

I456 

w 

G 

F 

*i 

6 

I3.50 

44 

'• 

tt 

.612 

2 

1457 

w 

G 

G 

" 

6 

" 

4< 

44 

44 

.611 

2 

1458 

w 

I 

I 

a| 

6 

M.25 

.1 

r  tt 

•4 

.610 

2 

1459 

w 

H 

H 

** 

6 

44 

11 

44 

44 

.608 

2 

1460 

X 

I 

I 

ai 

6 

15.00 

it 

44 

44 

.617 

2 

1461 

X 

J 

I 

" 

6 

" 

ti 

44 

44 

.618 

2 

1462 

V 

F 

F 

«i 

5 

I3.I3 

it 

44 

it  - 

.630 

2 

I463 

w 

F 

F 

lk 

5 

14 

44 

44 

•* 

.608 

2 

1464 

V 

E 

E 

ai 

6 

12-75 

41 

tt 

iA*  i* 

.624 

2 

14^5 

V 

D 

E 

" 

6 

44 

" 

44 

44 

.623 

2 

1466 

w 

G 

G 

aj 

6 

I3-50 

it 

44 

44 

.613 

2 

1467 

w 

N,V 

G 

" 

6 

" 

" 

" 

" 

.606 

2 

TABULATION  OF  RIVETED  JOINTS. 


619 


RIVETED    BUTT-JOINTS— Continued. 

STEEL   PLATE— Continutd. 


*» 

Maximum  Stress  on  Joint  per 

c  c 
._  4, 

Sectional  Area 

3 

Square  Inch. 

^ 

O  w 

of  Plate. 

h*/i  O* 

"o 

>~"c3 

Bearing 

Shear- 

Sf*> 

»—  > 

Ofc, 

at    ,_ 

Surface 
of 

ing 
Area  of 

4)  u 

t/3<O 

§&g 

!•§« 

0 

"o 
>» 

>-  £ 
22 

Rivets. 

Rivets. 

-Sj3 

§irt«j 

§          * 

lra-3* 

c  «5 

c 

AJ 

w  w    . 

Gross. 

Net. 

||c 

«8§  « 

gO-sS 

'»  «  §5 

P.**  i-  > 

js 

i 

I.  si 

h 

H 

u 

C/5 

W 

H 

sq.  in 

sq.  in. 

sq.in. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

7.270 

4-°43 

3-227 

11.928 

58000 

36140 

65000 

81430 

22030 

62.3 

7-215 

3.968 

3-247 

11.928 

58000 

44490 

81050 

99040 

26960 

76.7 

500 

6.193 

3.538 

2-655 

9.940 

59050 

36670 

64190 

85540 

22850 

62  .0 

— 

6.232 

3.560 

2.672 

9.940 

59050 

36200 

63370 

84420 

22690 

61.2 

6.632 

3.921 

2.711 

9.940 

60000 

4243° 

71610 

103580 

28250 

70.5 

600 

6.632 

3-921 

2.711 

9-940 

60000 

38720 

65440 

94730 

25840 

64.5 

6.965 

4-243 

2.722 

9.940 

58000 

46630 

76550 

119320 

32680 

80.3 

500 

6.974 

4.246 

2.728 

9.940 

58000 

38900 

63890 

99400 

27290 

67.0 

7.230 

4-519 

2.711 

9.940 

58000 

39180 

62690 

104500 

28500 

67-5 

200 

7.250 

4.528 

2.722 

9.940 

58000 

40360 

65060 

108220 

29630 

70.0 

7-564 

4-837 

2.717 

9-940 

57410 

38570 

60450 

107610 

29410 

67.2 

7-565 

4-843 

2.722 

9.940 

57410 

3^160 

61170 

108830 

29800 

68.2 

6.986 

3.726 

3.260 

7.216 

55000 

3375° 

63280 

72330 

32670 

60.  i 

7.020 

3-744 

3.276 

7.216 

55000 

3453° 

64740 

74000 

3359° 

62.7 

7-393 

4.158 

3-234 

7.216 

55000 

36760 

65340 

84010 

37650 

66.0 

7.488 

4.212 

3.276 

7.216 

55000 

35120 

62440 

80280 

36440 

63-8 

7.918 

4-658 

3.260 

7.216 

55000 

41930 

71270 

101840 

46010 

76.2 

300 

7-956 

4.680 

3.276 

7.216 

55000 

36800 

62560 

89370 

40570 

66.9 

8.241 

5-039 

3.202 

7.216 

57290 

39320 

64290 

101  180 

44900 

68.6 

400 

8.249 

5.042 

3-207 

7.216 

57290 

36850 

60290 

94790 

42130 

64-3 

600 

7.488 

3-744 

3-744 

9-425 

55000 

32080 

64150 

64150 

25480 

58-3 

7.440 

3.720 

3.720 

9-425 

55000 

32060 

64110 

64110 

25300 

58.3 

7-931 

4.199 

.3-732 

9-425 

55000 

34120 

64440 

72510 

28710 

60.0 

7.880 

4.172 

3.708 

9-425 

55000 

34000 

64220 

72250 

28420 

61.8 

8.262 

4-59° 

3.672 

9-425 

57290 

36490 

65680 

82110 

32000 

63.6 

8.249 

4-S83 

3.666 

9-425 

57290 

36020 

64830 

81040 

31520 

62.8 

8  662 

5.002 

3.660 

9-425 

57290 

37720 

65310 

89260 

38490 

65.8 

8.664 

5.016 

3-648 

9-425 

57290 

37540 

64850 

89170 

345'o 

65-5 

9-255 

5-553 

S-?02 

9-425 

55940 

37300 

62160 

9325° 

36630 

66.6 

9.282 

5-574 

3.708 

9-425 

55940 

37000 

61610 

92620 

36440 

66.1 

8.259 

5.109 

3-I50 

7-854 

55000 

3578o 

57840 

93810 

37620 

65.0 

7-965 

4-925 

3  040 

7-854 

57290 

36960 

59770         96840 

37500 

64-5 

7-950 

3.738 

4.212 

11.928 

55000 

31000 

.66130        58690 

20720 

56-5 



7-949 

3-744 

4-205 

ii  .928 

55000 

31090 

66020        58780 

20720 

56-5 



8.269 

4-I3I 

4-138 

i  i  .  928 

57290 

33*5° 

66350        66240 

22980 

57-8 

8.181 

4.090 

4.091 

11.928 

55940 

33240 

66250 

66240 

22720 

58.0 

620 


APPLIED    MECHANICS. 


TABULATION    OF   SINGLED 
STEEL   PLATE—  Continued. 


No. 

of 
Test. 

Sheet  Letters. 

Pitch. 

No. 
of 
Rivets. 

Width 
of 
Joint. 

Nominal 
Thickness. 

Size  of 
Rivets 
and 
Holes. 

Actual 
Thick- 
ness of 
Plate. 

Lap. 

Plate 

Covers. 

Plate. 

Covers. 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

1468       W 

H 

I 

at 

6 

M-25 

5/8 

3/8 

iA  *  XB 

-613 

2 

1469       W 

I 

N&" 

" 

6 

" 

44 

44 

•* 

.609 

2 

1470       X 

J 

I 

2i 

6 

15.00 

44 

44 

44 

.619 

2 

1471        X 

J 

I 

" 

6 

44 

44 

44 

ii 

.616 

2 

1472       V 

K 

at 

5 

13-13 

44 

44 

it 

.628 

2 

1473  :    W 

F 

F 

44 

5 

" 

ii 

44 

it 

.609 

2 

i474  i    W 

H 

D 

at 

5 

13-75 

44 

41 

it 

.609 

2 

1475  ;  w 

G 



5 

1 

.610 

2 

1476  ;    W 

I 

I 

2* 

5 

14.38 

it 

44 

44 

.610 

2 

1477       W 

I 

I 

41 

5 

44 

ii 

44 

44 

.6oy 

2 

1478       X 

I 

J 

3 

5 

15.00 

44 

44 

44 

.616 

2 

1479       X 

I 

J 

14 

5 

44 

44 

44 

44 

.623 

2 

1480       G 

E 

E 

3* 

4 

12.50 

44 

44 

44 

.625 

2 

1481        H 

E 

E 

•» 

4 

" 

44 

44 

44 

.621 

2 

1482        Z 

K 

K 

2 

6 

12.00 

3/4 

7/16 

tt*» 

•736 

2 

1483 

Z 

K 

K 

" 

6 

44 

44 

44 

41 

•757 

2 

!  1484       Z 

K 

K 

a* 

6 

12.75 

44 

44 

ii 

.742 

2 

1485 

Z 

P 

P 

K 

6 

44 

44 

44 

" 

.762 

2 

1486 

Z 

L 

M 

2* 

6 

I3-50 

44 

11 

ii 

•749 

2 

1487 

z 

L 

M 

44 

6 

44 

11 

*4 

it 

.764 

2 

14  8 

z 

N 

N 

2f 

6 

I4-25 

44 

ii 

ii 

•745 

2 

1489 

z 

'o 

N 

" 

6 

" 

44 

44 

it 

•735 

2 

1490 

z 

K 

2i 

6 

12.75 

44 

44 

»A  *  i\ 

•723 

2 

1491 

z 

P 

P 

" 

6 

" 

44 

ti 

44 

752 

2 

1492 

z 

M 

L 

ai 

6 

I3-50 

44 

44 

44 

•736 

2 

1493 

z 

M 

M 

44 

6 

44 

44 

44 

it 

•754 

2 

1494  j     Z 

N 

0 

at 

6 

14.25 

44 

44 

44 

.760 

2 

1     ~ 
H95        z 

0 

0 

44 

6 

44 

44 

11 

44 

.760 

2       i 

1496 

Y 

Q 

P 

a* 

6 

15.00 

44 

44 

it 

•745 

2 

M97 

Y 

P 

P 

44 

6 

44 

44 

44 

44 

•725 

2 

1498 

Z 

L 

L 

at 

5 

I3-I3 

II 

ii 

44 

•733 

2 

1499 

z 

L 

L 

44 

5 

44 

44 

44 

44 

•744 

2 

1500 

z 

N 

M 

2* 

5 

13-75 

44 

44 

it 

.762 

2 

1501 

z 

N 

N 

44 

5 

44 

44 

it 

44 

•727 

2 

1502 

z 

O 

P 

at 

5 

14.38 

II 

4< 

44 

.722 

2 

1503 

z 

O 

O 

" 

5 

II 

.741 

2 

TABULATION  OF  RIVETED  JOINTS. 


621 


RIVETED    BUTT-JOINTS— Continued. 
STEEL   PLATE— Continued. 


Sectional  Area 
of  Plate. 

Bearing 
Surface 
of 
Rivets. 

Shear- 
ing 
Area  of 
Rivets. 

Tensile 
Str'gth 
of  Plate 
per 
Sq.  In. 

Max.  Stress  on  Joint  per  Sq.  In. 

Efficiency  of 
Joint. 

Temperature 
of  Joint  in 
Deg.  Fahr^ 

gill 

l«15 

£  g</>0 

c-  cS 

o  w  o  j« 

•afc-gcL 
£§£•0 

§bCU« 

.  c  o  <u 

cx'C<£2  > 

sg^S 

Sfflu-g 

be 

C        w 

'£-      oj 

$*0> 

%     * 

Gross. 

Net. 

sq.  in. 

sq.  in. 

sq.  in. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

3-735 

4-597 

4.138 

11.928 

572>o 

34260 

65110 

72330 

25090 

59-8 

8.690 

4-579 

4.  in 

11.928 

" 

34790 

66030 

73540 

25350 

60.7 

9.285 

5-i°7 

4.178 

ii  .928 

55940 

34980 

63600 

77740 

27230 

62.5 

9.240 

5.082 

4.158 

11.928 

" 

34770 

63220 

77270 

26930 

62.1 

8.239 

4.706 

3-533 

9.940 

55°oo 

36350 

63640 

84770 

30130 

66.1 

7.978 

4-552 

3.426 

9.940 

57290 

37100 

65020 

86400 

29780 

64-7 

8.362 

4-936 

3.426 

9-940 

" 

38150 

64630 

93110 

32090 

66.5 

8  381 

4-950 

3-43i 

9-94° 

" 

38120 

64540 

93120 

32140 

66.5 

8.833 

5.402 

3-431 

9.940 

" 

38620 

63140 

99410 

343io 

67.4 

8-739 

5-3'3 

3.426 

9.940 

" 

38180 

62830 

97430 

3358o 

66.6 

9.240 

5-775 

3-465 

9.940 

55940 

38480 

61570 

102630 

35770 

68.7 

9-345 

5.841 

3-504 

9.940 

" 

38410 

61430 

102440 

36110 

68.6 

7-8i3 

5.000 

2.813 

7-952 

55000 

37340 

58360 

103730 

36690 

67.9 

7-763 

4.968 

2-795 

7-952 

" 

38440 

60060 

106760 

37520 

69-9 

8.847 

4-431 

4.416 

9-425 

59000 

31990 

63870 

64090 

30030 

54-2 

9.099 

4-557 

4-542 

9.425 

" 

31980     63860 

64070 

30870 

54-2 

9-475 

5-023 

4-452 

9.425 

" 

34440     64960 

73920 

34620 

58.3 

9-723 

5-*5i 

4.572 

9  425 

41 

34700     67340 

73790 

35800 

58.8 

IO.II2 

5.618 

4-494 

9-42S 

14 

35000       63000 

78750 

37550 

59-3 

10.329 

5-745 

4-584 

9-425 

" 

36780  I  66130 

82870 

40310* 

62.3 

10.624 

6-154 

4.470 

9-425 

" 

38120     65810 

90600 

42970* 

64.6 

10.488 

6.078 

4.410 

9-425 

11 

34000       58670 

80860 

37830 

57-6 

9-233 

4-353 

4.880 

11.928 

" 

31050     65860 

58750 

24030 

52.6 

9.596 

4.520 

5.076 

11.928 

•4 

32000  \  67940 

65340 

25740 

54-2 

9  .951 

4-983 

4.968 

ii  .928 

" 

34270     68430 

68640 

28590 

58-0 

|    10.179 

5  082 

5-090 

11.928 

" 

33770  |  67540 

67520 

28810 

57-2 

10  845 

5-7*5 

5-130 

11.928 

•» 

34900      66230 

73780 

31730 

59-1 

10.838 

5.708 

5-!30 

ii  .928 

" 

35810     67990 

75650 

3254° 

60.6 

«i-*75 

6.146 

5-029 

11.928 

60420 

38470 

69940 

85480 

36040 

63.6 

10.  890 

5-996 

4.894 

11.928 

" 

37740     68650 

83980 

34460 

62.4 

9.624 

5-501 

4-I23 

9.940 

59000 

35000     61230 

81700 

33890 

57-° 

9  776 

5-572 

4.204 

10.030 

" 

36470     63990 

84810 

35550 

61.7 

10.478 

6.  192 

4.286 

9.940 

11 

38760     65590 

94750 

40850* 

65-7 

10.004 

5-9!5 

4.089 

9.940 

It 

36740      62130 

89880 

36970 

62.2 

10.390 

6  329 

4.061 

9-940 

" 

3793° 

62270 

97050 

39650* 

64-3 

10.663 

6-495 

4.168  |     9.940 

II 

40630 

65810 

90600 

42970* 

68.8 

*  Steel  rivets. 


622 


APPLIED    MECHANICS. 


TABULATION    OF   SINGLE- 
STEEL  PLATE— Continued. 


No. 

of 
Test. 

Sheet  Letters. 

Pitch. 

No. 
of 
Rivets. 

Width 
of 
Joint. 

Nominal 
Thickness. 

Size  of 
Rivets 
and 
Holes. 

Actual 
Thick- 
ness of 
Plate. 

Lap. 

Plate. 

Covers. 

Plate. 

Covers. 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

Jb°4 

Z 

M 

L 

a* 

6 

13-50 

3/4 

7/16 

iA*il 

.722 

2 

*505 

Z 

M 

M 

" 

6 

11 

•• 

" 

44 

.762 

2 

1506 

Y 

N 

0 

at 

6 

14-25 

11 

" 

44 

.727 

2 

15°7 

Y 

N 

0 

" 

6 

" 

11 

it 

44 

•735 

2 

1508 

Y 

Q 

Q 

a* 

6 

15.00 

" 

" 

44 

•737 

2 

T5°9 

Y 

Q 

Q 

" 

6 

" 

" 

" 

K 

•753 

2 

1510 

Y 

L 

L 

8| 

5 

13-13 

" 

" 

44 

.748 

2 

'5" 

Y 

L 

L 

" 

5 

" 

" 

" 

(1 

•755 

2 

1512 

Z 

.... 

at 

5 

13-75 

" 

" 

44 

.750 

2 

1513 

Z 

N 

N 

" 

5 

" 

44 

" 

41 

.764 

2 

JSH 

Y 

o 

0 

3| 

5 

M-iS 

" 

" 

" 

.760 

2 

1515 

Y 

0 

0 

K 

5 

" 

" 

" 

44 

.746 

2 

*5'6 



PS 

P 

3 

5 

15.00 

" 

11 

44 

•749 

2 

1517 

Y 

P; 

Q 

" 

S 

" 

" 

" 

" 

.741 

2 

1518 

Z 

K 

K 

3t 

4 

12.50 

14 

" 

44 

•756 

2 

1519 

Z 

K 

K 

" 

4 

41 

11 

it 

44 

•  741 

2 

1520 

Z 

K 

K 

3i 

4 

13.00 

it 

41 

ii 

-763 

2 

1521 

Z 

K 

K 

" 

4 

11 

" 

41 

41 

.718 

2 

1522 

Z 

M 

M 

3t 

4 

13-50 

11 

44 

(I 

.742 

2 

1523 

Z 

M 

M 

" 

4 

" 

" 

M 

•754 

2 

TABULATION   OF  RIVETED  JOINTS 


623 


RIVETED    BUTT-JOINTS—  Continued. 

STEEL   PLATE  -Continued. 


Sectional  Area 
of  Plate. 

Bearing 
Surface 
of 
Rivets. 

Shear- 
ing 
Area  of 
Rivets. 

Tensile 
Str'gth 
of  Plate 
per 
Sq.  In. 

Max.  Stress  on  Joint  per  Sq.  In. 

Efficiency  of 
Joint. 

Ill 
pi 

%      u 

§£§« 

|o|s 

£  cc^ 

£  OC/2>0 

o  b£«  2 

.  C   U   4J 

Ills 

c3ffi"^ 

U> 
C       en 

|o| 

Gross. 

Net. 

sq.  in. 

sq.  in.  '• 

sq.  in. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

\ 

0.761 

4-346  - 

5-415 

14.726 

59000 

29090 

65350 

52460 

19280 

49-3 

10.287 

4  572 

5-7I5 

14.726 

59000 

30010 

67520 

54010 

20960 

50.8 

10.367 

4.914 

5-453 

14-726 

60420 

33610 

70900 

f389o 

23660 

55-6 

10.474 

4.961 

5-513 

14.726 

60420 

33660 

71070 

63960 

23940 

55-7 

i  i  .  070 

5-542 

5-528 

14  726 

60420 

3478o 

69470 

69650 

26140 

57-5 

11.310 

5-662 

5-648 

14.726 

60420 

34670 

69250 

69420 

26620 

57-4 

9  918 

5-243 

5-675 

12.272 

60420 

36120 

68380 

76680 

29210 

59-7 

9.928 

5.209 

4.719 

12.272 

60420 

36940 

70400 

77710 

29880 

61.1 

10.328 

5  •  64° 

4.688 

12.272 

59000 

33730 

61770 

74320 

28390 

57.0 

10  505 

5-73° 

4-775 

12.272 

59coo 

35260 

64640 

77570 

30100 

59-7 

10.929 

6.179 

4-75° 

12.272 

60420 

3793° 

67080 

87260 

36220 

62.7 

io-735 

6.072 

4.663 

12.272 

60420 

38720 

68460 

89150 

33870 

64.0 

T  i  .  205 

6.524 

4.681 

12.272 

55520 

36530 

62740 

87440 

36610 

65.8 

11.108 

6.477 

4-631 

12.272 

60430 

38740 

66440 

92920 

35060 

64.1 

9-465 

5.685 

3.780 

9.818 

59000 

3756o 

62360 

9378o 

36110 

63  6 

9-2/7 
9-934 

5-572 
6.  119 

3-705 

9.818 
9.818 

59000 
59000 

39000 
37600 

64930 
61040 

97650 
97900 

36850* 
38040* 

66.1 
63-7 

9  348 

5.753 

3  590 

9.818 

59000 

36000 

58440 

9374° 

34280 

61  .0 

10.032 

6.322 

3.710 

9.818 

59000 

40040 

63540 

108270 

40^10* 

67-7 

10  .  187 

6.417 

3.770 

9.818 

59000 

39720 

63050 

107320 

41210* 

67-3 

*  Steel  rivets. 


624 


APPLIED  MECHANICS. 


TABLE 

TABLE  OF  EFFICIENCIES  OP 


STEEL  PLATE. 


Plate. 

No.  of 
Test. 

Pitch  of  Rivets. 

if" 

if" 

it" 

2" 

*" 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

per  cent. 

1308 

67-5 

68.9 

72.8 



1313 

68.3 

69.9 

72.1 



t"    .    .    .    .- 

1314 
1323 
1324 

65-9 
64.8 

66.4 
67.1 
62.6 

68.3 
68.6 
64.4 

I°'3 
67-5 

1337 

63-9 

65-9 

68.2 

1338 

.... 

64.8 

61.7 

I35S 

58.7 

60.6 

1356 

74-6 

73.3 

1359 

72.7 

72.0 

.... 

1360 

68.2 

69.0 

70.0 

70.9 

I"   .    .    .    .  - 

^67 
1368 

68.3 

70.1 
64-5 

70.3 
61.7 

68.5 

64.8 

1379 

63.0 

65.2 

67.4 

1380 





58.7 

59-6 

1395 

.... 

57-3 

59-5 

300 

300 

1396 

.... 

66.1 

79.6 

64.7 

.... 

1401 

68.1 

68.6 

350 
68.9 

- 

1402 

.... 

64-5 

66.3 

400 

76.0 

w 

1411 

.... 

250 
66.9 

300 
76.4 

66.3 

1412 

.... 

.... 

.... 

260 
62.8 

59-2 

1425 

... 

.... 

.... 

60.  1 

680°i 

300 

1426 

.... 

.... 

.... 

.... 

59-3 

, 

1443 

.... 

.... 

.... 

51-5 

1444 

.... 

.... 

60.  1 

66.0 

300 

76.2 

«// 

M5i 

62.7 

63.8 

66.9 

.    .    .    . 

1452 

.   .  . 

58.3 

60.0 

1463 

58-3 

61.8 

1464 

... 

.. 

. 

56.5 

1481 

56-5 

1482 

.   .  . 

, 

54-2 

58.3 

1489 

.       . 

54-2 

58.8 

1490 
1503 

.   .  . 

.... 

52.6 
54-2 

1523 

'       ' 

.... 

NOTES. — Figures  in  heavy- face  type  denote  that 
Super  numbers  state  the  temperature  of 


TABULATION  OF  RIVETED  JOINTS. 


62  5 


IsO.  i. 

SINGLE-RIVETED  BUTT-JOINTS. 


STEEL  PLATE. 


Pitch  of  Rivets. 

Diam- 
eter of 
Rivet- 
holes. 

*" 

a|" 

2*" 

af" 

2J// 

at" 

3" 

3*" 

3*" 

3*" 

per  ct. 

per  ct. 

per  ct. 

per  ct. 

per  ct. 

per  ct. 

per  ct. 

per  ct. 

p«r  ct. 

per  ct. 

in. 

i 

I 

i 
i 

f 
* 

J 
i 

f    i 
* 
i 
i 
it 
* 

1 
i 

t 

i 

Ii 

72.7 
74.0 
70.1 
70.0 
61.5 
63-4 

70.7 

72.3 
63-7 
68.4 

76.8 
67.1 
72.1 
68.5 

69.8 
68.3 
6s-4 
68.4 

77.1 
75-o 

73-5 
74-  * 

m 

70.7 
68.2 
65.4 
64.1 

72.4 
70.4 
65-3 
68.3 

75-5 
75  6 
69.8 
69.5 

67:; 
68.3 

6S!8 
69.8 

73-2 
75-° 

66.4 
too 

77.6 

a  oo 
60.3 

61.5 

360 

67.2 

56.5 

6480°6 

64°03 
63.6 
62.8 
57-8 
58.0 

59.3 

62.3 
58.0 
57  2 
49-3 
50.8 

756.°5 
66.7 

400 

72.8 

64-5 
700 

62.7 

58.0 

600 

77-3 
66.9 
62.3 

500 

76.7 

67.2 

400 
76.9 

62.O 
6l.2 

600 

81.5 

67.6 

800 
70-5 

64.5 

600 

80.3 
67.0 

200 

67-5 

70.0 

67.2 
68.2 

.... 

.... 

65*8 
65-5 
59-8 
60.7 

64.6 
57.6 

59-1 
60.6 
55  6 
55  7 

66.6 
66.1 

62.5 
62.1 

63"6 
62.4 

57-5 
'  57-4 

65.0 
64.5 

66.1 
64.7 

67.0 

61.7 

59-7 
61.1 

6^5 
66.5 

67  .'i 

66.6 

68.7 
68.6 

67^9 
69.9 

"± 

!;;; 

65~7 

62.2 

57.0 
59  7 

64'  3 

68.8 
62.7 
64.0 

65-8 
64.1 

63.6 

66.1 

ai 

67.7 
67.3 

joint  did  not  fracture  along-  line  ol  riveting, 
'oints  tested  at  temperatures  above  atmospheric. 


626 


APPLIED   MECHANICS. 


TABLE   NO.   2. 

TABLE  OF  DIFFERENCES  BETWEEN  THE  EFFICIENCIES  AND  RATIOS  OF  NET  TO 
GROSS  AREAS. — SINGLE-RIVETED  BUTT-JOINTS,  STEEL  PLATE. 


Plate. 

No.  of 
Test. 

Width  of  Plate  between  Rivet  Holes. 

Diameter 
of  Rivet 
Holes. 

i" 

ii" 

ii" 

if" 

*T 

if" 

if" 

i*" 

2" 

in. 
i    - 

r 
i 

r 

*  - 

i  - 
r 
* 

in. 

1308 
1313 
i3H 
1323 
i324 
1337 
1338 
1355 

1356 
1359 
1360 
1367 
1368 
1379 
1380 
1395 

1396 
1401 
1402 
1411 
1412 
H25 
7426 
1443 

1444 

i45i 
1452 
1463 
1464 
1481 

1482 
1489 
1490 
I5Q3 
1504 
1521 

perct. 

6.0 
6.8 
8.8 

K 

,fl 

18.8 

13- 
11. 
ii. 
ii. 
ii. 
9-7 
6.8 
7-3 

200 
9-0 

II.  I 

II  .  I 
250 
13-5 
850 
12.8 

10.  I 

300 
12.2 

4.6 

6.4 

9-4 
8-3 
8-3 
9-5 
9-4 

4-1 
4-1 

5-5 
7-1 
4.8 
6.4 

perct. 

4-6 
5-6 
6-3 
7.0 
8.1 
9.6 
8.8 
7.6 

9.1 
7.7 

9.0 

IO.  I 

5-4 

I'9 
6.7 

6.6 

300 
19.6 

8.6 

IO.O 

300 

2O.  I 

6.2 
300 
15-2 
360 
I7.2 

6-5 

9.7 

7-5 
7-1 
8.9 
7-8 
8.0 

\\ 

7-9 
7.2 

8.2 

8-3 

per  ct. 

6.1 

5-4 
7.2 
6.1 
8.6 
9-4 
5-9 
7-8 

per  ct. 

per  ct. 

perct. 

per  ct. 

perct. 

per  ct. 

in. 

i 

i 

i 
i 

i 

* 

i 
i 

i 

* 
i 
I 
i 
i 
ii 
ii 

1 

i 

i 

3 

i 
i 

i! 

4.0 
5-6 
xi.  7 

8.9 
5-8 
10.5 

87:i 

7.6 

9-2 
12.  I 

8-5 

ii.  8 

2.1 

3-5 
6.6 

i:i 

13-4 

"•3 

8.  '2 
8.8 

'i:? 

7-5 

7.8 

11 

9-9 
5-9 

2.  I 

350 
6.4 

400 
17.2 

7-4 

200 

4-7 
5-9 

700 
10.2 

5-3 

300 

17  4 

8.1 
8.0 
7-2 
7.2 
8.0 

3.7 

6.6 
6.4 

7-9 

8 

6.2 

3.8 

9-6 
7-i 
7-4 
10.4 

9.4 

7-2 

9.8 
9-5 

10.5 
10.6 

i:? 

5.2 

6.2 

7-9 
9.8 

.... 

5-3 

500 
16.4 

400 
14-9 

6-5 
6-9 

600 
21.7 

400 

7-5 

600 
3-2 

8.1 
7-6 
7-5 
7-1 

67 
-0.4 

8.6 

1:1 

8.6 

ll!03 
3.5 

500 
I7.2 

6.9 
4-9 
4-i 

5-2 

400 
14.9 

600 
II.4 

5-4 

500 

17.8 

4.0 

500 

19.4 

6.1 

.... 

200 

5-° 
7-5 

3.2 
4.2 



6.6 
6.0 

9.0 

7-6 

3.1 
2.7' 

7-5 
7-4 

6.2 

5.8 

6.2 

6.1 

3.9 
5.9 

-.02 

4-7 
2.4 

5-i 

"6.6 
3.1 

6.2 

7-4 

3-4 
7-9 

7.6 

5.8 

3.5 

6.0 

2.1 

-0.6 

2i" 

perct. 
4.7 
4.3 

NOTES. — Figures  in  heavy-faced  type  denote  that  joint  did  not  fracture  along  line  of  riveting 
Super  numbers  state  the  temperature    of   joints    tested   at   temperatures    above 
atmospheric. 


TABULATION  OF  RIVETED  JOINTS. 


627 


TABLE   NO.  3. 

EXCESS  IN  STRENGTH  OF  NET  SECTION  IN  JOINT  OVER  STRENGTH  OF  TENSILE 
TEST-STRIP. — SINGLE-RIVETED  BUTT-JOINTS,  STEEL  PLATE. 


Plate. 

in. 
i 

t 
* 

t   • 

i 

A 

No.  of 
tTest. 

Width  of  Plate  between  Rivet  Holes. 

Diameter 
of  Rivet 
Holes. 

x" 

ii" 

if 

ij" 

if" 

2" 

1308 
1313 

y*3*4 

1323 
1324 
1337 

1338 

1355 
1356 

1359 
1360 

1367 
1368 
1379 
1380 
1395 

1396 

1401 
1402 
1411 
1412 
1425 

1426 

1443 
1444 
1451 

1452 

M63 
1464 
1481 

1482 
1489 
1490 
1503 
1504 
1521 

3er  ct. 

9-8 
n.  i 
!5-4 
13-4 
17.4 
19.7 
29.6 
17.6 

21.2 

18.1 

19-3 
19-5 
20.9 
18.2 
15-5 
14.6 

200 

15.6 

T5.8 

20.9 
260 
25-2 
350 
25.5 

20.3 
300 
2O.  O 

9.8 

IS-I 
17.7 

16.6 
16.6 

20.2 
20.0 

8-3 

8.2 

ii.  6 
15.2 
10.8 
14.4 

per  ct. 

7-2 

8.8 
10.5 
11.7 
14.4 
17.0 
16.5 
14-5 
14.1 
11.9 
15.0 
16.8 
9-7 
15-9 

12.6 

12.5 

300 

32-8 

14.4 

17.8 

300 

35-8 
ii.  8 
2880°8 

360 

34-4 
13.2 

18.8 

i3-5 
17.2 
16.8 
15.8 
18.4 

10.  I 

14.1 

16.0 
14-5 
17-4 
17.6 

perct. 

9.1 
8.1 

9-2 

9-7 
M.7 
15-9 
10.6 
14.2 

12.0 
12-5 
10.  1 

13-5 
17.8 
10.6 

3-4 

360 
10.3 

400 

28.9 

12.8 
200 

5-5 
10.8 

700 
I2.3 

10.  1 

300 

20.  6 

13.7 

14-6 
13.2 
13-6 
•15-3 
6.8 

12.  1 
I2.3 
15-2 
I5.0 

I4.6 

perct. 

perct. 

perct. 

perct. 

per  ct. 

perct. 

10.6 
10.1 

in. 
f 

* 

! 

i 
i 

i 

t 

* 

X 
X 

i 
1 

* 
i 

X 

Ii 

It 

* 
i 

t 

i 
i 

Ii 

:i 

6.2 

8-5 
14.7 
14.6 

10.  0 

18.0 

9-i 
ii  .0 
11.9 
14.4 

20.1 
I4.I 

ie.'i 

1:1 

10.7 

4-7 
2-5 

21.0 

17.7 

12.7 
13-5 

9.6 
5.8 
15-6 
11.7 

12.8 

17.9 

14.9 
II.4 
I6.3 
15-9 

16.1 
16.5 
8.9 
10.3 

8.2 

9-8 

12.2 
IS-* 

'.'..'. 

8.7 

600 
27.0 

400 

25.8 

"•3 

12.  I 
600 

39-7 

400 
12.2 

600 
5-2 
14.0 
I3.2 

!3-7 
13.0 

11.5 
-0.6 

15.8 

13-6 
13.2 
16.5 

196.°5 
5.6 

600 
28.7 

"•3 

8.7 

7-3 

8.6 

400 

24.0 

600 
19.4 

9-i 

500 
28.1 

6.4 
600 
32.0 

10.  1 

a8°°i 

12.2 

5.3 
6.5 

11.1 
10.1 

15-7 
13-5 

5.2 
4.3 

12.8 

12.7 

10.2 

9-7 

lo.'i 

9.8 

e.'i 

9.2 

3.8 

8.5 
4-7 
9.6 

II.  2 

5.3 

II.  0 

13-3 

5-5 
"•5 
13.0 
10.0 

5.7 

10.  1 

4i 

2i" 

per  ct. 
7.7 
6.9 

verage  of 
all  joints. 

16.2 

14.4 

12.3 

12.9 

11.9 

II.  0 

9-7 

ii.  8 

7.0 

Norms.— Figures  in  heavy-faced  type  denote  that  joint  did  not  fracture  along  line  of  riveting. 
Super   numbers  state  the  temperature  of    joints  tested    at    temperatures    abcrc 
atmospheric. 


628 


APPLIED    MECHANICS. 


In  the  Report  of  Tests  made  at  Watertown  Arsenal  during  the  fiscal  year 
ended  June  30,  1891,  is  the  following  account  of  another  series  of  tests  on  riveted 
joints: 

44  Comprised  in  the  present  report  are  113  tests  made  with  steel  plates  of 
1/4",  5/16",  3/8",  and  7/16"  thickness  with  iron  rivets  machine  driven  in  drilled 
or  punched  holes. 

44  The  plates  used  were  from  material  used  in  earlier  tests,  the  results  of 
which  have  been  published  in  previous  reports. 

41  In  the  use  of  metal  once  before  tested,  such  plates  were  selected  as  had 
not  been  overstrained  previously,  or  those  in  which  the  elastic  limit  had  been 
but  very  slightly  exceeded. 


SINGLE-RIVETED 


STEEL  PLATE. 


Sheet  Letters. 

Nominal 
Thickness. 

Size 

Actual 

No.  of 
Test. 

Pitch. 

No.  of 
Rivets. 

Width 
of 
Joint. 

and 
Kind 
of 

Thick- 
ness 
of 

Lap. 

Plate. 

Covers. 

Plate. 

Covers. 

Holes. 

Plate. 

49i3 

H 

C 

D 

in. 

2* 

5 

in. 
i3-72 

in. 

i/4 

in. 
3/i6 

in. 

7/8    d 

in. 
•  247 

in. 

2 

49M 

L 

D 

D 

" 

5 

13.69 

" 

" 

"    " 

.248 

" 

49'5 

M 

E 

E 

8* 

5 

14.32 

i« 

it 

.4        tt 

.247 

44 

4916 

M 

D 

D 

it 

5 

14-33 

11 

" 

tt        tl 

.247 

44 

49i7 

M 

E 

E 

3 

5 

15.00 

11 

tt 

tt        tt 

.246 

14 

4918 

M 

E 

E 

" 

5 

14.98 

it 

it 

tt        It 

.247 

11 

4985 

Q 

D 

D 

3* 

4 

14.00 

A 

" 

I          " 

•  3°9 

I 

4987 

s 

C 

" 

4 

14.01 

" 

14 

"        " 

.310 

li 

499* 

Q 

1 

" 

4 

14.05 

« 

" 

it        U 

-308 

If 

5«5 

R 

A 

A 

I 

10 

IO.02 

5/'6 

11 

1/2     " 

.306 

I* 

5"6 

R 

A 

.... 

I* 

8 

10.02 

" 

tt 

tt       It 

-3°4 

" 

5127 

R 

B 

I* 

7 

10.51 

" 

" 

tt       tt 

.310 

" 

5143 

L 

P 

2 

7 

14.03 

7/16 

5/i6 

7/8    P- 

.440 

I* 

5M4 

L 

O 

0 

" 

7 

14.01 

it 

11 

"    d. 

.440 

" 

SMS 

0 

0 

Q 

2* 

6 

I3-50 

44 

44 

it    tt 

-434 

" 

5H6 

M 

O 

— 

" 

6 

I3-5I 

u 

it 

"    P- 

.421 

14 

5M7 

O 

P 

P 

2* 

6 

15-02 

" 

44 

41    d. 

•413 

44 

5148 

N 

P 

.... 

14 

6 

15.02 

" 

H 

"    P- 

.411 

" 

SiSS 

K 

S 

.... 

2* 

5 

13.75 

41 

It 

M    d. 

•425 

TABULATION  OF  RIVETED  JOINTS. 


629 


"The  present  tests  are  supplementary  to  those  of  earlier  reports,  and  occupy 
a  place  intermediate  between  the  elementary  forms  of  joints  and  the  more  elab- 
orate types  of  joints  which  have  been  investigated. 

"  Wide  variation  has  been  given  the  pitches,  and  rivets  of  extreme  diameters 
have  been  used  for  the  purpose  of  including  joints  in  which  these  features  have 
been  carried  to  their  extreme  limits. 

"  The  efficiencies  of  the  joints  are  stated  in  per  cent  of  strength  of  the  solid 
plate." 


BUTT-JOINTS. 


STEEL  PLATE. 


Sectional  Area 
of  Plate. 

Bearing 

Shear- 
ing 

Tensile 
Str'gth 

Maximum  Stress  on  Joint  per  Sq.  In. 

Effi- 

Surface 
of 

Area 
of 

OI 

Plate 

Tension 
on 

Tension 

on  Net 

Compres- 
sion on 

Shearing 

ciency 
of 

Gross. 

Net. 

Rivets. 

Rivets. 

per 
Sq.  In. 

Gross 
Section 

Section 
of 

Bearing 
Surface 

of 
Rivets. 

Joint. 

of  Plate. 

Plate. 

of  Rivets. 

sq.  in. 

sq.  in. 

sq.  in. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

3-39 

2.31 

i.  08 

6.01 

59180 

44180 

65760 

140650 

25270 

75-7 

3-40 

2.31 

1.09 

6.01 

61470 

435oo 

64030 

135690 

24610 

70.8 

3-54 

2.46 

i.  08 

6.01 

58170 

46300 

66630 

151780 

27270 

79-6 

3-54 

2.46 

i.  08 

6.01 

58170 

435oo 

62590 

142570 

25620 

74.8 

3-69 

2.61 

i.  08 

6.01 

58170 

46290 

65440 

158150 

28420 

79.6 

3-70 

2.62 

i.  08 

6.01 

58170 

44400 

62700 

152110 

27330 

76.3 

4-33 

3-°9 

1.24 

6.28 

56760 

24040 

33690 

83950 

16580 

42-3 

4-34 

3.10 

1.23 

6.28 

57000 

26770 

3748o 

93710 

18500 

46.9 

4-33 

3.10 

1.23 

6.28 

56760 

33940 

47410 

119500 

23400 

59-8 

3.07 

i.  54 

i-53 

3-92 

61130 

35930 

71620 

72090 

28140 

58.8 

3-°5 

1.83 

1.22 

3-i4 

61130 

41280 

68800 

103200 

40100 

67-5 

3.26 

2.17 

I.OQ 

2.74 

61130 

39250 

58960 

117380 

46690 

64.2 

6.17 

3.38 

2.79 

8.41 

59390 

32540 

59410 

71970 

23880 

54-8 

6.16 

3.48 

2.69 

8.41 

59390 

23360 

41350 

53490 

17110 

30.  -\ 

5-86 

3-58 

2.28 

7.21 

52910 

42250 

60160 

108600 

34340 

79.8 

5-69 

3-4° 

2.29 

7.21 

61650 

39740 

66500 

98730 

31360 

64.5 

6.20 

4-03 

2.17 

7.21 

52910 

44150 

67920 

126130 

37960 

83-4 

6.17 

3-94 

2.23 

7.21 

61650 

36660 

57410 

101430 

3*370 

60.0 

5-84 

3.98 

1.86 

6.01 

59000 

40270 

59100         126450 

39130 

68.3 

630 


APPLIED    MECHANICS. 


TABULATION   OF  SINGLE- 
STEEL  PLATE. 


No.  of 
Test. 

Sheet  Letters. 

Pitch. 

No.  of 
Rivets. 

Width 
of 
Joint. 

Nominal 
Thickness. 

Size  and 
Kind 
of 
Holes. 

Actual 
Thick- 
ness   o: 
Plate. 

Lap. 

Plate. 

Plate. 

Plate. 

Plate. 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

4933 

I 

J 

H 

5 

10.62 

i/4 

i/4 

id. 

.252 

2 

4934 

J 

J 

" 

5 

10.65 

" 

" 

41 

•253 

2 

4939 

L 

K 

4 

4 

11.50 

" 

" 

i   d. 

.250 

2 

494°* 

J 

J 

it 

4 

11.50 

0 

ti 

44 

.256 

2 

4941 

K 

J 

44 

4 

ii  Si 

(i 

44 

iid. 

.252 

2 

4942 

K 

J 

" 

4 

11.52 

" 

ti 

41 

.250 

2 

4943 

K 

K 

(l 

4 

11.50 

K 

44 

iid. 

.252 

2 

4944 

E 

J 

K 

4 

ii  .50 

" 

" 

44 

•253 

2 

4945 

K 

K 

3i 

4 

12.52 

" 

" 

it 

.248 

2 

4946 

L 

G 

44 

4 

12.55 

t> 

44 

ii 

•253 

2 

4947* 

N 

H 

44 

4 

13-52 

(i 

" 

44 

.247 

2 

4948* 

N 

H 

" 

4 

13-52 

" 

it 

ii 

•247 

2 

4949 

M 

L 

3* 

4 

14-51 

44 

ii 

ii 

.248 

2 

495°* 

M 

L 

44 

4 

14-51 

" 

it 

44 

.247 

2 

4961 

B 

E 

i* 

6 

10.52 

3/8 

3/8 

id. 

.388 

2 

4979 

E 

E 

2f 

5 

11.84 

44 

44 

i   d. 

.384 

2 

5131 

K 

K 

I± 

8 

12.  OO 

7/16 

7/16 

Id. 

•427 

i-75 

5132 

N 

O 

44 

8 

12.00 

ii 

ii 

IP. 

•415 

i-7S 

5133* 

K 

K 

If 

8 

13.00 

«< 

ii 

|d. 

.427 

i-75 

5134 

N 

N 

" 

8 

13.00 

" 

it 

IP- 

•4*3 

i-75 

5135 

M 

M 

I* 

8 

M-03 

" 

ii 

Id. 

.422 

i-75 

5T36 

" 

8 

13-99 

" 

ii 

IP- 

.420 

i-75 

5137 

L 

M 

3 

7 

14.02 

44 

" 

Id. 

•4«4 

1-75 

5138 

P 

M 

" 

7 

14.05 

" 

ii 

IP- 

.420 

i-75 

5139 

O 

K 

II 

6 

12.06 

<« 

ii 

iid. 

.428 

2 

5140 

M 

M 

2t 

6 

14.28 

" 

4* 

it 

.421 

2 

5Mi* 

L 

L 

2} 

5 

13-73 

" 

4* 

M 

.438 

2 

5142* 

Q 

0 

3* 

5 

15-67 

44 

it 

M 

.422 

2 

*  Pulled  off  rivet-heads. 
t  Pulled  off  3  rivet-heads. 
$  Pulled  off  2  rivet-heads. 


TABULATION  OF  RIVETED  JOINTS. 


631 


RIVETED   LAP-JOINTS. 


STEEL  PLATE. 


Sectional  Area 
of  Plate. 

Bear- 
ing 
Surface 
of 
Rivets. 

Shear- 
ing 
Area 
of 
Rivets. 

Tensile 
Strength 
of  Plate 
per 
Sq.  In. 

Maximum  Stress  on  Joint  per  Sq.  In. 

Effi- 
ciency 
of 
Joint. 

Tension 
on  Gross 
Section 
of  Plate. 

Tension 
on  Net 
Section 
of  Plate. 

Comp.  on 
Bearing 
Surface 
of  Rivets. 

Shear- 
ing of 
Rivets. 

Gross. 

Net. 

sq.  in. 

sq.  in. 

sq.  in. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

2.68 

i-57 

I.IO 

3-00 

61000 

39750 

67850 

96840 

35510 

65.1 

2.70 

i-59 

i.  ii 

3-00 

61000 

39660 

67360 

96490 

35700 

65.0 

2.87 

1.87 

1.  00 

3-M 

58150 

40560 

62250 

116400 

37070 

69.7 

2.94 

1.92 

1.02 

3-i4 

61000 

37010 

56670 

106670 

34650 

60.6 

2.90 

1.77 

1-13 

3-98 

61000 

43280 

70900 

111060 

31530 

70-9 

2.88 

i-75 

1.13 

3.98 

58150 

42770 

73880 

119010 

30950 

73-5 

2.90 

1.64 

1.26 

4.91 

58150 

41130 

72730 

94660 

24290 

70.7 

2.91 

1.64 

1.27 

4.91 

58150 

40200 

71330 

92110 

23820 

69.1 

3.10 

1.86 

1.24 

4.91 

58150 

40030 

66720 

100080 

25270 

69.1 

3-i7 

i.  91 

I    26 

4.91 

61470 

41770 

69320 

105080 

26970 

68.0 

3-34 

2.10 

1.24 

4.91 

55740 

42240 

67180 

112970 

28730 

75-7 

3-34 

2.10 

1.24 

4.91 

59180 

42600 

67760 

114760 

28980 

71.9 

3-6o 

2.36 

1.24 

4.91 

61470 

41390 

63140 

120180 

30350 

67.3 

3.58 

2-35 

1.23 

4.91 

58170 

42150 

64210 

122680 

30730 

72.4 

4.08 

2-33 

J-7S 

2.65 

5834° 

25950 

45440 

60500 

39950 

44.4 

4-55 

2.63 

1.92 

3-93 

58340 

33050 

57I9° 

78330 

38270 

56.6 

5-12 

2.13 

2-99 

4.81 

59000 

31740 

76290 

54350 

33780 

53-8 

4  99 

1.98 

3.01 

4.81 

52910 

27820 

70100 

46110 

28860 

52.6 

5-55 

2.56 

2.99 

4.81 

59000 

31100 

67420 

57730 

35880 

52-7 

5-37 

2-37 

2-99 

4.81 

61140 

30370 

68820 

54550 

339*0 

49-7 

5-90 

2-95 

2.95 

4-81 

61650 

29240 

58490 

58490 

35870 

47-4 

5-89 

2-95 

2-94 

4.81 



31870 

63630 

63840 

39020 

5-94 

3-35 

2.60 

4.21 

5939° 

27580 

48900 

63000 

38910 

46.4 

5-9° 

3-24 

2.66 

4.21 

52910 

28530 

51940 

63270 

39980 

53-9 

5-16 

i-95 

3-21 

7.36 

58090 

28190 

74610 

45320 

19770 

48-5 

6.01 

2.85 

3.16 

7.36 

61650 

34850 

73490 

66280 

28460 

56.5 

6.01 

3.28 

2.74 

6.14 

59390 

3356o 

61490 

73610 

32850 

56.5 

6.61 

3-97 

2.64 

6.14 

56960 

30420 

50650 

76170 

32750 

53-4 

632. 


APPLIED    MECHANICS. 


TABULATION    OF    DOUBLE- 
CHAIN-RIVETING-STEEL  PLATE. 


•«  o 

No. 
of 
Test 

Sheet  Letters. 

Pitch. 

mce  Apart  o 
ws,  Centre  t 
ntre. 

Total 
Num- 
ber o 
Riv- 
ets. 

Width 
of 
Joint. 

Nominal 
Thickness. 

Size 
and 
Kind  o 
Holes. 

Actual 
Thick- 
ness o 
Plate. 

Lap. 

Plate. 

Covers. 

*»  o  v 

Plate 

Covers. 

Q 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

4911 

K 

C 

C 

*f 

2i 

10 

13.10 

i/4 

3/i6 

5/8  d. 

•253 

JT! 

4912 

L 

C 

C 

" 

" 

IO 

13.10 

44 

44 

44 

•253 

14 

49i9 

L 

E 

D 

af 

2* 

10 

14-32 

44 

K 

7/8  d. 

.247 

i}| 

4920 

L 

E 

D 

" 

44 

10 

14-32 

44 

44 

" 

.249 

" 

4921 

K 

C 

B 

3* 

44 

8 

12.52 

44 

44 

44 

.252 

« 

4922 

K 

C 

C 

44 

44 

8 

12.49 

it 

44 

" 

.252 

" 

4923 

J 

A 

A 

3* 

44 

6 

II-S7 

44 

44 

44 

•257 

44 

4924 

J 

A 

44 

44 

6 

"•53 

44 

44 

44 

•255 

11 

4925 

L 

B 



4* 

44 

6 

13.09 

'« 

41 

44 

.251 

»• 

4926 

K 

C 

B 

44 

41 

6 

13.  10 

*' 

44 

44 

.230 

" 

5128 

R 

/"v 

C 

C 

I* 

2 

M 

12.27 

5/i6 

3/16 

1/2  d. 

•  3°4 

if 

5I29 
5130 

Q 
S 

D 

*i 

» 

12 

I3-58 

.« 

u 

.. 

•305 
•  307 

« 

4993 

Q 

E 



3* 

I* 

S 

14.05 

44 

44 

i  d. 

•309 

i$ 

4995 

Q 

E 

.... 

11 

I* 

8 

14.06 

44 

44 

44 

•3°5 

i* 

4997 

Q 

E 



44 

2 

8 

14.08 

44 

K 

44 

.308 

tj 

495i 

B 

I 

I 

2* 

2* 

8 

8.52 

3/8 

1/4 

3/4  d. 

•392 

i* 

4952 

E 

R 

44 

" 

8 

8.51 

41 

5/6 

44 

-383 

" 

4953 

E 

R 

af 

44 

8 

10.51 

44 

•• 

it 

.388 

" 

4954 

E 

N 

H 

fci 

44 

8 

10.03 

44 

i/4 

44 

384 

" 

4955 

E 

S 

3* 

44 

8 

12.50 

44 

5/16 

•« 

•383 

« 

4957 

H 

0 

O 

3t 

44 

8 

I4-5I 

" 

44 

44 

-369 

" 

49.S8 

H 

M 

M 

4' 

44 

8 

I4-52 

44 

i/4 

44 

•369 

" 

4959 

B 

S 

S 

4* 

44 

6 

12.42 

*« 

5/i6 

44 

.388 

it 

4960 

E 

L 

N 

i4 

44 

6 

12.42 

44 

i/4 

44 

•384 

» 

4967 

C 

M 

M 

at 

2* 

10 

14-38 

44 

" 

i  d. 

•375 

2 

4969 

E 

M 

3l 

44 

8 

13-50 

" 

4' 

44 

.382 

« 

4970 

F 

— 

44 

44 

8 

13-58 

44 

5/i  6 

44 

.380 

" 

4971 

J 

P 

P 

3* 

K 

8 

15.46 

44 

44 

44 

•379 

" 

4973 

E 

S 

S 

4t 

44 

6 

13-50 

44 

44 

44 

.385 

« 

4975 

I 

O 

P 

4f 

44 

6 

14-65 

44 

44 

« 

•373 

» 

4977 

J 

P 

P 

5f 

44 

6 

16.08 

" 

44 

•• 

•379 

K 

4956 

K 

N 

N 

3* 

a* 

8 

12.48 

7/16 

1/4 

3/4  d. 

.427 

!J 

4968 

N 

0 

— 

** 

2* 

10 

14.41 

5/i  6 

id. 

.409 

2 

TABULATION  OF  RIVETED  JOINTS. 


633 


RIVETED    BUTT-JOINTS. 

CHAIN-RIVETING—STEEL  PLATE. 


Sectional  Area 
of  Plate. 

Bear- 
ing 
Surface 
of 
Rivets. 

Shear- 
ing 
Area 
of 
Rivets. 

Tensile 
Strength 
of 
Plate 
per 
Square 
Inch. 

Maximum  Stress  on  Joint  per  Sq.  In. 

Effi- 
ciency 
of 
Joint. 

Gross. 

Net. 

Tension 
on 
Gross 
Section 
of 
Plate. 

i  Tension 
on 
Net 
Section 
of 
Plate. 

Compres 
sion  on 
Bearing 
Surface 
of 
Rivets. 

Shearing 
on 
Rivets. 

sq.  in. 

sq.  in. 

sq.  in. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

3-3i 

2.52 

1.58 

6.14 

58150 

49090 

64480 

102850 

26470 

84-4 

3-3i 

2.52 

1.58 

6.14 

61470 

51960 

68250 

108860 

28010 

84.5 

3*54 

2.46 

2.16 

12.02 

61470 

46810 

67370 

76720 

1379° 

76.1 

3-57 

2.48 

2.18 

12.02 

61470 

457°o 

65790 

74840 

13570 

74-3 

3-16 

2.27 

1.76 

9.62 

58150 

46330 

64490 

83180 

15220 

79.6 

3-'5 

2.27 

1-76 

9.62 

58150 

46730 

64850 

83640 

15300 

80.3 

2.97 

2.30 

r-35 

7.2I 

61000 

49520 

63940 

108930 

20400 

81.2 

2.94 

2.27 

J-34 

7.21 

61000 

49460 

64050 

108510 

20170 

8i.x 

3-2Q 

2.63 

1.32 

7.21 

61470 

51440 

64350 

128210 

23470 

83-7 

3-oi 

2.41 

1.  21 

7  21 

58150 

55500 

69320 

138070 

23170 

95-4 

3-73 

2.66 

2.13 

5-49 

61130 

46690 

66650 

83240 

32300 

76.4 

4.27 

3-20 

2.14 

5-49 

56760 

49040 

65430 

97850 

38140 

86.4 

4-15 

3-23 

1.84 

4.70 

57000 

46480 

59720 

104840 

41040 

81.5 

4-34 

3-" 

2-47 

12.57 

56760 

44740 

62430 

78600 

iS45o 

78.8 

4.29 

3-°7 

2.44 

12.57 

56760 

45490 

63570 

7998o 

15530 

80.  i 

4-34 

3-io 

2.46 

12.57 

56760 

45530 

63740 

80330 

15720 

80.2 

3-34 

2.16 

2-35 

7.07 

5973° 

43290 

66940 

61520 

20450 

72.4 

3-26 

2.  II 

2.30 

7.07 

58340 

42380 

65470 

60070 

19540 

72.6 

4.08 

2.9t 

2-33 

7.07 

58340 

46590 

65330 

81590 

26890 

79.8 

3-85 

2.70 

2-30 

7.07 

58340 

49130 

70060 

82240 

26750 

84.2 

4-79 

3-64 

2.30 

7.07 

58340 

48610 

63970 

101250 

32940 

83-3 

5-35 

4.25 

2.21 

7.07 

56670 

48500 

61060 

117420 

36700 

85.5 

5.36 

4-25 

2.21 

7.07 

56670 

47700 

60160 

115700 

36170 

84-1 

4.82 

3-95 

r-75 

5.30 

59730 

43070 

52560 

118630 

39170 

72.1 

4-77 

3-91 

i-73 

5-30 

58340 

42520 

51870 

117230 

38260 

72.8 

5-39 

3-52 

3-75 

I5-7I 

57870 

42890 

65680 

61650 

14720 

74-  i 

5.16 

3-63 

3-o6 

12-57 

58340 

44263 

62920 

74640 

18170 

75-9 

5.16 

3-64 

3-°4 

".57 

54290 

43240 

61290 

73390 

17750 

79-6 

5.86 

4-34 

3-03 

".57 

5713° 

44910 

60650 

86860 

20940 

78.6 

5.20 

4.04 

2.31 

9.42 

58340 

45980 

59180 

103510 

25380 

78.8 

5-46 

4-35 

2.24 

9.42 

59030 

46720 

58640 

113880 

27080 

79.1 

6.09 

4.96 

2.27 

9.42 

5713° 

44650 

54830 

119800 

28870 

78.1 

5-33 

4-05 

2.56 

7.07 

59000 

48120 

63300 

100190 

36280 

83-3 

5-89 

3.85 

4.09 

I5-7I 

61140 

433°° 

66340 

62440 

16260 

70.9 

634 


APPLIED    MECHANICS. 


TABULATION    OF    RIVETED 
DOUBLE-RIVETED   LAP-JOINTS. 


09 

c 

£  c 

Nominal 

8 

1 

Q 

Letters  ol 
>and  Cov 

part  of  Ro 
ang.  to  1 
vets. 

Rivets  in 
t  Row. 

Rivets  in 
id  Row. 

Rivets  in 
d  Row. 

a 

'o 

o 

Thickness. 

"o 
•o 

C 

"5  « 

i 

o 

oJ« 

JS 

rt.  tfS 

s-,  C 

'o  8 

o;s 

-8 

u 

i2 

u 

§1 

is 

1 

j«E 

o 

.2  rt'o 

dfc 

o'c/ii 

ciH 

•a 

> 

o 

SE 

t)  O 

a 

rt 

in 

(X 

° 

£ 

fc 

* 

OH 

U 

c/5 

^ 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

in. 

4935 

J-J 

2i 

2t 

5 

5 

10.68 

1/4 

7/8  d. 

•257 

1ii 

4936 

I-J 

4* 

44 

5 

5 

10.53 

44 

44          44 

•251 

k4 

4937 

L-M 

2i 

** 

5 

5 

14.38 

" 

44         44 

.248 

4938 

I-L 

u 

** 

5 

5 

14.40 

44 

44         44 

.249 

44 

4999 
5000 

4963 
4965 

ft? 

E-E 
E-E 

3* 

2 

2i 

4 

i 

6 

4 

4 
6 
6 

14.02 
14.00 
10.50 

12.  OO 

5/,(6 
3/8 

I         " 
I         P. 

3/4  <!• 

•  305 
.306 
-387 
-384 

2 

4! 

498i 
4983 
5149 

E-D 
D-H 

M-L 

2t 
2* 
2 

ft 

5 
5 
7 

5 
5 
7 

11.83 
I4-36 
14.00 

7/,6 

7/8  " 

•385 
•  370 
-425 

2 

5*5° 

M-M 

44 

7 

7 

I4.OO 

'*     P- 

•423 

44 

K-K 

21 

44 

6 

6 

JT  .  ^7 

k* 

"    d. 

.428 

«4 

5I52 

L-0 

4- 

44 

6 

6 

*3-5° 

"    P. 

.440 

,| 

5153 

0-0 

2* 

" 

6 

6 

15.01 

44 

"    d. 

.409 

Ij7B 

5154 

O-O 

K-K 

2t 

*' 

6 
5, 

6 
5 

15.02 

" 

"  t 

.412 
.422 

44 

DOUBLE-RIVETED    BUTT-JOINTS. 


4927 

MDE 

2* 

2t 

5 

4 

I4-36 

i/4 

3/16 

7/8  d. 

•  250 

'.H 

4928 

LDE 

5 

4 

I4-35 

.247 

4929 

KBC 

H 

44 

4 

3 

12.51 

44 

4 

•255 

" 

493° 

KG 

" 

4 

3 

12.50 

44 

' 

.251 

44 

493i 
4932 

HDD 
HCC 

?,* 

" 

3 
3 

2 

2 

13-12 
13-12 

" 

4 

.248 
.246 

44 

DOUBLE-RIVETED   LAP-JOINTS. 


5"9 
5120 

0-0 

p-p 

# 

»t 

4 

4 

3 
3 

14.00 
14.03 

S/i6 

d. 
P- 

•303 
•305 

i* 

5»2I 

R-R 

" 

It 

4 

3 

14.03 

" 

d. 

.302 

i* 

5122 

O-O 

" 

4 

3 

14.03 

14 

P- 

.304 

" 

5123 

O-O 

44 

2 

4 

3 

14.02 

44 

d. 

.302 

it 

5124 

P-P 

4 

3 

14.02 

P- 

.307 

•4v 

TREBLE-RIVETED    LAP-JOINTS 


5157 

KK 

2t 

a| 

5 

5 

5 

iS-M 

7/16 

7/8  d. 

'432 

i« 

5158 

OP 

3 

5 

5 

5 

!5-05 

44 

"     4t 

.412 

5159 

5100 

PP 
LL 

3 

" 

4 
4 

4 

4 

4 
4 

12.78 
I3-50 

tt 

o     11 

'432 
.438 

ii 

TABULATION  OF  RIVETED  JOINTS. 


635 


JOINTS.— STEEL   PLATE. 


CHAIN-RIVETING. 


Sectional  Area 

0) 

•<-. 

•2 

Maximum  Stress  on  Joint  per  Sq.  In. 

c 

of  Plate. 

J| 

1 

ft. 

"5 

o  £ 

been 

Tension 

Tension 

Compres- 

>> 

Gross. 

Net. 

b£  ^ 

.£5 

II 

nj  _rt  G 
~  pl  *"* 

on  Gross 
Section  ot 

on  Net 
Section 

sion  on 
Bearing 

Shearing 
on 

c 

V 

2~o 

£  "o  c/}          Plate 

of  Plate  . 

Surface 

Rivets. 

sg 

CQ 

C/3 

H 

of  Rivets. 

H 

sq.  in. 

sq.  in. 

sq.  in. 

sq.  in. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

Ibs. 

2-74 

1.62 

2.25 

6.01 

61000 

42770 

72350 

52090 

19500 

70.1 

2.64 

1.54 

2.  2O 

6.01 

62300 

42350 

72600 

58180 

18600 

i     67.9 

3-57 

2.49 

2.17 

6.01 

61470  '       47870 

68630 

78760 

28440 

77.0 

3-58 
4.28 

2.49 
3-06 

2.l8 
2.44 

6.01 
6.28 

61470         48530 
56760  :        46070 

69780 
64440 

79700 
80820 

28910 
31400 

78.9 
80.  i 

4.28 

3-°3 

2-53 

6.28 

593°o   .        43900 

62100 

74370 

29960 

74.1 

4-06 

2.32 

3.48 

5-30 

58340   |        40570 

70900 

47330 

69.5 

4.61 

2.88 

3.46 

5-30 

58340   i        42150 

67470 

56160 

36660 

72.2 

4-55 

2.63 

3-85 

7-85 

53730   !        38790 

67100 

45840 

22480 

72.2 

5-31 

3-46 

3-70 

7.85           56670           4}c5o 

66070 

61780 

29120 

76.0 

5-95 

3-35 

5-21 

8.41        ;       61650              40620 

72150 

4fil-9o 

28740 

i     65.8 

5-93 

3-24 

8.41           61650          379'o 

69380 

41940 

26730 

:      61.4 

5-79 

3-54 

4.49 

7.21 

59000           43150 

70570 

73  -1 

5-94 

3-55 

4.78 

7.21           59390          38870 

65040 

48310 

12O2O 

!     65.4 

6.14 

3-99 

4.29 

7.21                 52910                43530 

6f  QQO 

62310 

37070 

82.3 

6.  19 

5-81 

3-95 
3.96 

4.48 
3.69 

7-21         j        52910                40380 

6.01           59000          38850 

63290 

56990 

55800 

61170 

34670 
37550 

1     76.3 
;      65.8 

ZIGZAG-RIVETING. 


3  59 

•So 

•97 

10.82 

58170 

48150 

69140 

87740 

15980 

80.3 

3-54 

.46 

•95 

10.82 

61470 

47420 

68240 

86090 

15520 

77-i 

3-!9 

•3° 

•56 

8.41 

58150 

46610 

64650 

95320 

17680 

80.2 

3-'4 

.26 

.56 

8.41 

58150 

47520 

66020 

95640 

17740 

81.7 

3-25 

.60 

•05 

6.01 

59180 

47640 

59550 

M745° 

25760 

80.5 

3-23 

.58 

.08 

6.01 

59180 

46720 

5S490 

139720 

25110 

78.9 

ZIGZAG-RIVETING. 


4.24 

4.28 

4-23 

4.27 

3-03 
3.02 
3.02 
3  01 

.  12 
.20 
.IT 
•19 

5-50 

5  50 
5  50 
5-50 

56760 
5930° 
54350 
54350 

42750 
40630 
42990 
44^40 

59830 

57580 

60220 
6^000 

855^0 
79050 
86180 
86450 

32960 
31620 
33060 
34420 

75-3 
68.  S 
79.1 
81  6 

4-23 

3.02 

.  II 

5-5° 

54350 

44870 

62850 

89050 

345  10 

82.5 

4-33 

3-03 

.22 

5-50 

593°° 

43490 

62150 

84820 

31400 

73-3 

CHAIN-RIVETING. 


5-93 

6.20 

5-52 
5-91 

4.04 
4.40 
4.01 
4.38 

5-66 
5-4» 
4-54 
4.60 

9.02 
9.02 
7.21 
7.21 

59000 
52910 
58090 
59390 

4^720 
48710 
48040 
46430 

67100 
68630 
66130 

62650 

47900 
55820 
584- 
59650 

30060 
3348o 
36780 
38060 

77  5 
92.1 
82.7 
78.2 

636  APPLIED   MECHANICS. 

In  the  design  of  a  riveted  tension-joint  the  problem  usually 
presents  itself  in  the  following  form : 

Given,  in  all  particulars,  the  two  plates  to  be  united,  to 
design  the  joint  ;  i.e.,  to  determine,  i°,  the  diameter  of  rivet  to 
be  used  ;  2°,  the  spacing  of  the  rivets,  centre  to  centre  ;  and,  3°, 
the  lap. 

In  regard  to  the  determination  of  the  lap,  the  common 
practice  has  been  already  explained  and  very  little  has  been 
done  experimentally. 

In  order  to  determine  the  diameter  and  the  spacing  of  the 
rivets  by  the  usual  methods  of  calculation,  it  becomes  neces- 
sary to  know  the  three  following  kinds  of  resistance  of  the 
metals,  viz.: 

i°.  The  tensile  strength  per  square  inch  of  the  plate  along 
the  line  or  lines  of  rivet-holes ; 

2°.  The  shearing-strength  of  the  rivet  metal ; 

3°.  The  resistance  to  compression  on  the  bearing-surface  of 
either  plate  or  rivet. 

Hence  we  need  to  ascertain  what  the  tests  cited  show  in 
regard  to  these  three  quantities. 

Tension. — The  tensile  strength  of  the  plate  used  should, 
of  course,  be  determined  by  means  of  tests  made  on  specimens 
cut  from  it.  Further  than  this,  questions  arise  as  to  the 
excess  tenacity  due  to  the  grooved  specimen  form,  and  as  to 
any  injury  due  to  punching  when  the  holes  are  punched. 

The  excess  tenacity  is,  of  course,  greater  with  small  than 
with  large  spaces  between  the  rivet-holes  ;  hence,  inasmuch  as 
the  tendency  is  toward  the  use  of  large  rivets,  and,  conse- 
quently, large  pitches,  the  excess  tenacity  applicable  in  practi- 
cal cases  becomes  small,  and  would  be  better  disregarded  in 
the  design  of  most  riveted  joints.  In  cases  where  the  holes 
are  drilled,  therefore,  we  should  use  for  tensile  strength  per 
square  inch  of  the  plate  along  the  line  of  rivet-holes,  the  tensile 
strength  per  square  inch  of  the  plate  itself. 


COMPRESSION.  637 


The  better  and  more  ductile  the  plate  the  less  is  the 
injury  done  by  punching;  but,  while  more  or  less  punching  is 
done,  the  better  class  of  work  is  drilled.  A  study  of  the  results 
in  the  cases  of  punched  plates  will  show  approximately  what 
allowance  to  make  for  the  weakening  due  to  punching  different 
qualities  of  plate. 

Shearing". — A  study  of  the  results  of  the  government  tests 
show  that  it  is  fair  to  assume  the  shearing-strength  of  the 
wrought-iron  rivets  used,  to  be  about  38000  pounds  per  square 
inch,  which  is  about  two  thirds  of  the  tensile  strength  of  the 
same  rivet  metal. 

For  steel  rivets,  of  the  kinds  now  prescribed  in  most  spec- 
ifications, the  shearing-strength  appears  to  be  about  45000 
pounds  per  square  inch. 

Compression. — To  determine  what  we  should  estimate  as 
the  ultimate  compression  on  the  bearing-surface  is  a  more 
difficult  problem  ;  for  if  a  joint  fails  in  consequence  of  too 
great  compression  on  the  bearing-surface  the  cause  of  the 
failure  does  not  exhibit  itself  directly,  but  in  some  indirect 
manner — probably  by  decreasing  the  resisting  properties  of 
either  the  plate  or  the  rivets,  and  hence  by  causing  either  the 
joint  to  break  by  tearing  the  plate  or  by  shearing  either  the 
rivets  or  the  plate  in  front  of  the  rivets,  but  at  a  lower  load 
than  that  at  which  it  would  have  broken  had  the  compression 
not  been  excessive;  and  hence  when  such  breakage  occurs  it  is 
difficult  to  say  whether  it  is  due  to  excessive  compression  re- 
ducing the  tensile  or  the  shearing  strength,  or  whether  its  full 
tensile  or  shearing  strength  was  really  reached. 

Observe,  moreover,  that  in  the  tables  of  Government  tests 
the  heavy  numbers  in  the  column  marked  "  Compression  on 
the  bearing-surface  of  the  rivets  "  indicate  that  the  plate  broke 
out  in  front  of  the  rivets,  which  might  be  due  to  excessive 
compression  or  to  a  deficiency  of  lap. 

While  more  experiments  are  needed,  it  would  seem  proba- 
ble that  we  might  deduce  some  conclusions,  at  least,  of  a  gen- 
eral nature,  in  regard  to  the  ultimate  compression  by  a  study 


638  APPLIED   MECHANICS. 

of  the  relations  existing  between  the  compression  per  square 
inch  on  the  bearing-surface  at  fracture  and  the  efficiency  of 
the  joint  as  shown  by  the  Government  tests. 

For  this  purpose  the  following  diagrams  (see  pages  631  and 
632)  have  been  plotted,  with  the  efficiencies  as  abscissae  and 
the  compression  per  square  inch  on  the  bearing-surface  at 
fracture  as  ordinates.  If  similar  diagrams  were  plotted  with 
the  efficiencies  as  abscissae  and  the  ratio  of  the  compression  per 
square  inch  on  the  bearing-surface  at  fracture  to  the  tensile 
strength  of  the  plate  as  ordinates  the  character  of  the  diagrams 
would  be  substantially  the  same,  as  the  plates  used  in  the  tests 
were  all  of  mild  steel  of  approximately  the  same  quality,  and 
hence  the  difference  in  tensile  strength  of  different  samples 
was  not  great. 

A  study  of  these  diagrams  shows  that  in  the  case  of  the 
i-inch  plates  experiments  were  made  with  compressions  up  to 
about  158,000  pounds  per  square  inch,  but  that  the  highest 
compression  reached  with  any  other  thickness  of  plate  was 
about  120,000  pounds  per  square  inch. 

Inasmuch  as  Kennedy  advises  the  use  of  96,000  pounds  per 
square  inch,  and  as  this  is  higher  than  the  values  that  have 
been  customarily  advocated,  it  would  hardly  seem  wise  to 
adopt  a  much  higher  value  unless  the  tests  furnish  us  sufficient 
evidence  for  such  a  procedure.  Considering  the  facts  stated 
above,  and  also  the  fact  that  in  the  cases  of  the  double-riveted 
joints  some  of  the  highest  compressions  were  accompanied  by 
a  decrease  in  efficiency,  it  would  seem  best  to  limit  our  esti- 
mate of  the  ultimate  compression  on  the  bearing-surface  to 
from  90,000  to  100,000  pounds  per  square  inch  until  we  have 
further  light  on  the  subject  derived  from  experiment  ;  and  it  is 
not  at  all  improbable  that  when  we  do  obtain  further  light  we 
may  find  ourselves  warranted  in  using  a  somewhat  higher 
value. 

The  reasoning  which  leads  to  the  above  conclusion  is,  of 
course,  based  on  evidence  which  is  not  conclusive,  because  of 
the  lack  of  tests  with  higher  compressions  on  the  bearing  sur- 


COMPRESSION. 


6380 


APPLIED    MECHANICS. 


W/TH   TWO 

STEEL  PLJTE.  •$' 'STEEL  PLJTE. 


ilnTOMmiilllfflt 

///>  f\  s>  si    ::: 

;i;;;;J:i:;i;;:i;;!i;i;i;;N;;  /soooo 

•  {      • 

j  |j  J_[  |j_[_[  U'i  1  r  H  t  ini*  LL1J  li  1  1     /?/?/?/?/) 
::;;::;:!;:::::;::::        7W0 

70 


<ft7 


7<?         W         ^        /^ 

face,  with  plates  thicker  than  one  quarter  of  an  inch.  On  the 
other  hand,  the  quarter-inch  plates  show  higher  efficiencies 
with  compressions  above  100000  pounds  than  they  do  with 
compressions  of  100000  pounds  or  less,  and  the  author  knows 
of  tests  upon  riveted  joints  in  T7¥-inch  plates  which  tend  to 
show  that,  with  good  wrought-iron  rivets,  it  would  be  perfectly 
safe  to  use  a  considerably  larger  number  for  compression  on 
the  bearing-surface,  in  designing  riveted  joints — at  least 
Iioooo  pounds  per  square  inch,  and  probably  more. 


COMPRESSION. 


It  will  be  observed  that  no  reference  has  been  made  to  the 
friction,  and  it  is  safer  to  leave  this  out  of  account,  as  the  tests 
show  that  slipping  takes  place  at  all  loads,  and  as  there. is  no 
friction  at  the  time  of  fracture. 

By  far  the  greater  part  of  the  tests  at  Watertown  Arsenal 
were  made  with  wrought-iron  rivets  in  mild-steel  plates,  this 
being,  at  the  time,  the  most  usual  practice,  although  steel 
rivets  were  sometimes  used.  At  the  present  time,  notwith- 
standing the  fact  that  steel  long  ago  superseded  wrought- 
iron  for  boiler-plate,  and  that  it  has,  to-day,  superseded 
wrought-iron  for  structural  shapes,  as  I  beams,  channel-bars, 
angles,  etc.,  and  that  the  use  of  steel  rivets  has  become  very 
extensive,  nevertheless  a  great  many  still  adhere  to  the  use 
of  wrought-iron  rivets,  and  feel  more  confidence  in  them  than 
they  do  in  steel  rivets.  Whereas  the  use  of  wrought-iron 
rivets  had  been  practically  universal,  the  qualifications  for  a 
good  wrought-iron  rivet  metal  became  pretty  well  known,  and 
while  sometimes  specifications  were  drawn  up  giving  the 
requirements  of  the  rivet  metal  for  tensile  strength,  ductility, 
etc.,  which  of  course  would  vary  more  or  less,  nevertheless 
the  variations  would  not  be  large.  A  study  of  the  Watertown 
tests  shows  that  the  wrought-iron  rivet  metal  used  in  those 
tests  had  a  tensile  strength  of  from  about  52000  to  about 
59000  pounds  per  square  inch,  with  a  percentage  contraction 
of  area  at  fracture  of  from  about  30  to  about  45.  With  this 
metal  the  shearing  strength  per  square  inch  seems  to  be  about 
f  of  the  tensile  strength  per  square  inch.  Of  course  other 
tests  are  necessary  to  show  whether  the  metal  can  be  properly 
worked,  and  whether  it  is  red-short  or  not,  such  as  that  the 
metal  should  bend  double,  whether  cold  or  hot,  without  cracks, 
and  that  cracks  should  not  develop  when  the  shank  is  ham- 
mered down,  cold  or  hot,  to  a  length  considerably  less  than 
the  diameter. 

When  steel  rivets  were  first    used,  the  steel  employed  was 


638 d  APPLIED   MECHANICS. 

not  an  extremely  soft  steel,  as  shown  by  the  few  cases  of  steel 
rivets  included  in  the  Watertown  Arsenal  tests  already  quoted, 
where  the  shearing-strength  per  square  inch  varied  from  about 
50000  pounds  per  square  inch  up  to  as  high  a  figure  as  65000 
pounds  per  square  inch;  and  by  Kennedy's  tests,  where  he  ap- 
parently fixes  on  from  about  49000  to  about  54000  pounds  per 
square  inch  as  the  shearing-strength  of  steel  rivets. 

Now  it  would  seem  that  metal  with  these  shearing-strengths 
would  have  a  tensile  strength  per  square  inch  which  would  not 
warrant  us  in  classifying  it  as  very  soft  steel. 

On  the  other  hand,  it  is  evident  that  brittleness  should  not 
in  any  way  be  tolerated  in  rivet  metal,  and  hence  it  would  seem 
that  at  least  soft  steel  should  be  used  for  rivets. 

The  specifications  proposed  by  the  American  Society  for 
Testing  Materials  prescribe  for  tensile  strength  per  square  inch 
of  steel  for  structural  rivets  from  50000  to  60000  pounds  per 
square  inch,  and  for  boiler-rivets  from  45000  to  55000  pounds. 

While  the  number  of  tests  that  have  been  made  upon  joints 
constructed  with  steel  rivets  is  not  large,  the  shearing-strength 
of  such  steel  rivets  as  are  in  use  to-day  is  not  very  far  from  45000 
pounds  per  square  inch,  as  a  rule. 

The  number  of  tests  of  joints  constructed  with  steel  rivets  is 
not  sufficiently  large  to  warrant  drawing  from  them  definite 
conclusions  regarding  the  ultimate  compression  on  the  bearing 
surface  in  such  joints.  Meanwhile,  it  would  be  advisable  to  use 
for  it  the  same  values  as  are  suitable  in  the  case  of  joints  made 
with  steel  plates  and  wrought-iron  rivets. 

The  following  table  contains  the  joints  tested  at  Watertown 
Arsenal,  which  were  made  with  steel  plate  and  wrought-iron 
rivets,  and  in  which  the  plate  broke  out  in  front  of  the  rivet.  It 
is  evident  that  only  four  of  them,  viz.,  4915,  4916,  4917,  4918, 
failed  in  consequence  of  excessive  compression  on  the  bearing 
surface,  and  that  the  breaking  out  of  the  plate  in  the  other  cases 

was  due  to  insufficiency  of  lap.     The  calculated  -j  was  obtained 


WIRE   AND    WIRE   ROPE. 


639 


by  the  method  described  on  page  554,  assuming  ^  =  55000,  and 
)8  =  38000,  and  jc  =  96000. 


"3 
-C 

V 

o 

II  II 

'3 

•s! 

u 

1 

d  ,   A 

O  i   o 

t/j 

\i 

•d 

£ 

ji* 

Kind  of 

Joint. 

^ 

o 
«„ 

jj 

o 
e 

•^ 

a; 

"5 

M  3, 

01     1 

C  P 

E 

S\ 

liil 

«! 

S'S 

«3fe 

d  ' 

1 

IH 

1 

P 
5: 

CQ 

0 

•^(3 

a 

5  ftffl  J 

•5*0 

6 

2 

a 

OH 

£ 

^ 

J 

J> 

^3 

0 

—  113 

in 

5. 

II 

s. 

Ins. 

Ins. 

Ins. 

Lbs. 

7i8 

Single  lap 

Iron 

n 

lo 

P 

1 

f 

1.25 

795io 

Tore  and 

1.18 

i.SS 

sheared 

719 

' 

V 

P 

•25 

80200 

Tore 

.18 

.55 

4947 

* 

D 

.00 

i  i  2970 

.60 

.  75 

A 

4948 

D 

.  oo 

114760 

Tore 

.60 

-75 

4949 

' 

D 

.00 

120180 

.60 

•  77 

767 
1442 

Single  butt 

- 

P 

i 

•  25 

.00 

1.25 

2.00 

95210 
107610 

Tore 

.67 

.50 

.65 

1443 

• 

• 

. 

r 

5 

.00 

2  .  OO 

108830 

•  5° 

.  70 

49  1  5 

^ 

- 

D 

JL 

.  oo 

2  .  OO 

151780 

•  29 

•  93 

4916 

' 

- 

D 

: 

J- 

.00 

2.00 

142570 

•  29 

.89 

4917 
4918 

- 

: 

D 
D 

1 

ft 

.  oo 

.00 

2  .  OO 
2.00 

158150 
152110 

.  29 
•  29 

.96 
•  93 

4985 

r 

D 

1 

if 

A 

.00 

I  .  OO 

83950 

.  oo 

•  57 

4987 

[ 

D 

1 

A 

.25 

1  •  25 

937io 

•  25 

.63 

4991 

I 

D 

•J 

V 

3, 

•  75 

I     ~  ~ 

i  19500 

•  75 

•  77 

298 

Reinforced 
lap 

\ 

i 

D 

i 

i 

j      -15 

1       .00 

[l.!2 

67300 

Sheared 
rivets 

.19 

.66 

299 

\ 

* 

D 

1 

i 

J        .10 
1        .12 

[l.I2 

68040 

,, 

.19 

.67 

5121 

Double  lap 

I 

D 

1 

V 

•50 

86180 

•  5° 

.56 

5122 

1 

P 

1 

V 

•50 

86450 

•59 

§  234.  Wire  and  Wire  Rope. — It  is  well  known  that  the 
process  of  making  wire  by  cold  drawing  greatly  increases  the 
strength  of  the  metal.  Annealing,  on  the  other  hand,  decreases 
the  strength,  and  increases  the  ductility.  It  is  not  the  purpose 
of  this  article  to  discuss  the  various  qualities  of  wire  required  and 
used  for  different  purposes.  Hence,  inasmuch  as  results  of  tests 
of  wrought-iron,  and  of  steel  wire,  have  already  been  given,  there 
will  be  given  here  only  a  few  tests  of  hard-drawn,  of  semi-hard- 
drawn,  and  of  soft  copper  wire. 

Wire  rope. — Wire  rope  is  used  for  a  great  many  purpose:,  as 
in  suspension  bridges,  in  hoisting,  in  haulage,  in  the  transmission 
of  power,  etc. 

While  flat  wire  rope  is  used  for  some  purposes,  and  while 
wire  rope  made  of  parallel  wires  is  used  in  large  suspension 
bridges,  the  greater  part  is  made  by  twisting  a  number  of  wire 


640 


APPLIED   MECHANICS. 


HARD-DRAWN  COPPER  WIRE. 


SOFT  COPPER  WIRE. 


Diameter. 

Tensile 
Strength 

Elastic 
Limit  per 

Contrac- 
tion of 

per 
Sq.  In. 

Sq.  In. 

Area. 

Inches. 

Lbs. 

Lbs. 

Per  Cent. 

0.166 

53050 

37100 

0.138 

60350 

22800 

o-i35 

56300 

28150 

o-i34 

5I05° 

2,7140 

o.  105 

61800 

41000 

5r 

o.  105 

57100 

35000 

49 

o.  105 

58900 

34000 

33 

o  .  106 

60300 

36000 

42 

o.  106 

59500 

34000 

39 

0.086 

58170 

27870 

0.086 

58620 

29310 

o  .  08  > 

6  1  5  10 

2  T  7OO 

•*  1oVw 

o  083 

66536 

206^0 

w  •  w<kj  O 

0.083 

65060 

*y  wo 

37334 

0.083 

66536 

29630 



Diameter. 
Inches. 

Tensile 
Strength 
per 
Sq.  In. 

Lbs. 

Elastic 
Limit  per 
Sq.  In. 

Lbs. 

Contrac- 
tion of 
Area. 

Per  Cent. 

0.163 
o.  162 
o.  162 

35730 
35770 
36640 

13760 
I2QQO 

0.083 

IO5OO 

0.083 
0.081 

o  .080 
o  .080 

29500 
33200 
33100 

IJ2CO 

70 

45 

SEMI-HARD-DRAWN    COPPER  WIRE. 


o.  106 

44300 

30000 

60 

o.  106 

45100 

29000 

65 

0.106 

455oo 

29000 

55 

o.  106 

45100 

31000 

67 

o.  106 

44900 

30000 

64 

strands  around  a  central  core,  which  may  be  of  tarred  hemp,  or 
which  may  be,  itself,  a  wire  strand,  the  wire  strands  being  made 
of  wires  twisted  together. 

In  the  case  of  a  wire  core,  the  strength  of  the  rope  is  a  little 
greater,  but  the  resistance  to  wearing  is  less. J 

The  most  usual  number  of  strands  is  six,  each  strand  contain- 
ing seven,  eighteen,  or  nineteen  wires,  though  other  numbers  of 
wires  are  sometimes  used. 

The  strength  that  can  be  realized  in  practice  is  always  less 
than  the  strength  of  the  rope,  and  is  determined  by  the  method  of 
holding  the  ends,  as  the  junction  point  of  the  rope  and  the  holder 
is  the  weakest  point. 

The  usual  methods  of  holding  the  ends  are  as  follows:  splic- 
ing, as  in  the  case  of  the  transmission  of  power,  passing  the  rope 
around  a  pulley,  or  around  a  thimble,  fastening  it  in  a  socket,  or 
in  a  clamp. 

The  diameter  of  the  drum  or  sheave  around  which  a  rope 


WIRE   AND    WIRE  ROPE. 


641 


passes,  should  not  be  so  small  as  to  cause  too  much  stress  to  be 
exerted  upon  some  of  the  wires,  in  consequence  of  the  bending- 
moment  introduced  by  the  curvature. 

Inasmuch  as  it  may  be  a  matter  of  convenience  to  have  here 
some  tables  giving  the  strength  of  rope  as  claimed  by  some  makers, 
there  will  follow  here  two  tables  of  the  strength  of  different  sizes, 
as  given  by  the  Roebling  Company  for  their  rope. 

The  following  explanations  are  given  by  the  Roebling  Com- 
pany, about  the  quality  of  the  metal  used : 

Iron,  open-hearth  steel,  crucible  steel,  and  plough  steel  pos- 
sess qualities  which  cover  almost  every  demand  upon  the  material 
of  a  wire  rope.  Copper,  bronze,  etc.,  are,  however,  used  for  a 
few  special  purposes. 

The  strength  of  iron  wire  ranges  from  45000  to  100000  pounds 
per  square  inch;  open-hearth  steel,  from  50000  to  130000  pounds 

SEVEN-WIRE  ROPE. 
Composed  of  6  Strands  and  a  Hemp  Center,  7  Wires  to  the  Strand. 


Approximate  Breaking-strain  in  Tons  of 

2000  Lbs. 

Trade  No. 

Diameter 
in  Inches. 

Approxi- 
mate 
Circum- 
ference 

Weight 
per  Foot  in 
Pounds. 

Transmission  or 
Haulage  Rope. 

Extra 

in  Inches. 

Swedish 

Cast 

Strong 
Cast  Steel. 

Plough 
Steel. 

Iron. 

Steel. 

ii 

l| 

4f 

3-55 

34 

68 

79 

91 

12 

!§ 

4i 

3.00 

29 

58 

68 

78 

13 

Ji 

4 

2.45 

24 

48 

56 

64 

14 

I| 

3* 

2  .00 

20 

40 

46 

53 

15 

I 

3 

1-58 

16 

32 

37 

42 

16 

1 

ai 

I  .  20 

12 

24 

28 

32 

17 

f 

2i 

0.89 

9-3 

18.6 

21 

24 

18 

t* 

4 

0-75 

7-9 

15.8 

l8.4 

21 

19 

f 

2 

0.62 

6.6 

13.2 

I5-I 

17 

20 

A 

if 

0.50 

5-3 

10.6 

12-3 

14 

21 

j 

l£ 

°-39 

4.2 

8.4 

9.70 

II 

22 

iV 

l| 

0.30 

3-3 

6.6 

7-50 

8-55 

23 

1 

1* 

0.22 

2.4 

4-8 

5.58 

6-35 

24 

f 

I 

0-15 

i-7 

3-4 

3-88 

4-35 

25 

A 

4 

o.  125 

1.4 

2.8 

3.22 

3-65 

642 


APPLIED    MECHANICS. 


NINETEEN-WIRE  ROPE. 
Composed  of  6  Strands  and  a  Hemp  Center,  19  Wires  to  a  Strand. 


Approximate  Breaking-strain  in  Tons  of 

2000  Lbs. 

Approxi- 

Weight. 

Trade  No. 

Diameter 
in  Inches. 

mate 
Circum- 
ference 
in  Inches. 

per  Foot  in 
Pounds. 

Standard  Hoisting 
Rope. 

Extra 
Strong 
Cast  Steel. 

Plough 
Steel. 

Swedish 

Cast 

Iron. 

Steel. 



2| 

8f 

n-95 

— 

— 



305 

— 

2* 

7l 

9-85 

— 

— 

— 

254 

i 

7i 

8.00 

78 

156 

182 

208 

2 

2 

^i 

6.30 

62 

124 

144 

165 

3 

if 

si 

4-85 

48 

96 

112 

128 

4 

If 

5 

42 

84 

97 

ill 

5 

I* 

4f 

3-55 

36 

72 

84 

96 

5* 

if 

3.00 

31 

62 

72 

82 

6 

I* 

4 

2.45 

25 

5° 

58 

67 

7 

3* 

2  .00 

2  I 

42 

49 

56 

8 

I 

3 

I.58 

17 

34 

39 

44 

9 

1 

I  .20 

13 

26 

30 

34 

10 

f 

24 

0.89 

9-7 

19.4 

22 

25 

10* 

f 

2 

O.62 

6.8 

13-6 

15-8 

18 

A 

if 

0.50 

5-5 

II  .0 

12-7 

14-5 

lof 

o.  39 

4-4 

8.8 

10.  I 

11.4 

ioa 

A 

i* 

0.30 

3-4 

6.8 

7-8 

8.85 

lob 

| 

it 

0.  22 

2-5 

5-° 

5-78 

6-55 

IOC 

T^> 

i 

0.15 

i  .  7 

3-4 

4-05 

4-5° 

lod 

i 

f 

0.  10 

I  .2 

2  -4 

2.70 

3.00 

per  square  inch;  crucible  steel  from  130000  to  190000  pounds 
per  square  inch;  and  plough  steel  from  190000  to  350000  pounds 
per  square  inch.  Plough  steel  wire  is  made  from  a  high  grade  of 
crucible  cast-steel. 

§  235.  Other  Metals  and  Alloys. — Copper  is,  next  to  iron 
and  steel,  the  me'al  most  used  in  construction,  sometimes  in  the 
pure  state,  especially  in  the  form  of  sheets  or  wire,  but 
more  frequently  alloyed  with  tin  or  zinc;  those  metals  where 
the  tin  predominates  over  the  zinc  being  called  bronze,  and 
those  where  zinc  predominates  over  tin,  brass.  Copper  in  the 
pure  state  was  used  not  long  ago  for  the  fire-box  plates  of  loco- 


IRON  AND    STEEL    WIRE.  643 

motive  and  other  steam-boilers, "as  it  was  believed  to  stand  better 
the  great  strains  due  to  the  changes  of  temperature  that  come 
upon  these  plates,  than  iron  or  steel ;  but  now  steel  or  iron  has 
almost  entirely  superseded  it  for  this  purpose,  except  in  some 
cases  where  the  feed-water  is  very  impure,  and  where  the 
impurities  are  such  as  corrode  iron. 

The  alloys  of  copper,  tin,  and  zinc  which  are  used  most 
where  strength  and  toughness  are  needed,  are  those  where  the 
tin  predominates  over  the  zinc ;  and  the  composition,  mode  of 
manufacture,  and  resisting  properties  of  these  metals,  together 
with  the  effect  of  other  ingredients,  as  phosphorus,  have  been 
very  extensively  investigated  with  reference  to  their  use  as  a 
material  for  making  guns,  instead  of  cast-iron. 

Accounts  of  tests  made  on  these  alloys  will  be  found  as 
follows : — 

Major  Wade  :   Ordnance  Report,  1856. 

T.  J.  Rodman  :  Experiments  on  Metals  for  Cannon. 

Executive  Document  No.  23,  46th  Congress,  2d  session. 

Materials  of  Engineering  :  Thurston. 

No  attempt  will  be  made  to  give  a  complete  account  of  the 
results  of  these  tests ;  but  a  table  will  be  given  on  page  639  for 
convenience  of  use,  showing  rough  average  values  of  the  resist- 
ing powers  of  some  metals  and  alloys  other  than  iron. 

§  236.  Timber.  —  However  extensively  iron  and  steel  may 
have  superseded  timber  in  construction,  nevertheless,  there  are 
many  cases  in  which  iron  is  entirely  unsuitable,  and  where 
timber  is  the  only  material  that  will  answer  the  purpose ;  and 
in  many  cases  where  either  can  be  used,  timber  is  much  the 
cheaper.  Hence  it  follows  that  the  use  of  timber  in  construc- 
tion is  even  now,  and  as  it  seems  always  will  be,  a  very  impor- 
tant item. 

Another  advantage  possessed  by  timber  is,  that,  on  yielding, 
it  gives  more  warning  than  iron,  thus  affording  an  opportunity 
to  foresee  and  to  prevent  accident. 

If  we  make  a  section  across  any  of  the  exogenous  trees,  as 


644 


APPLIED   MECHANICS. 


Specific 
Gravity. 

Tensile 
Strength 
per  Sq.  In. 

Modulus 
of 
plasticity. 

Brass  cast     .          .               .          .... 

8.7C-6 

18000 

9I7OOOO 

49OOO 

14230000 

Bronze  unwrought  : 
84.29  copper  +  15.71  tin  (gun  metal) 
82.81       "      +  17.19  " 
81.10      "      +  18.90  " 
78.97       "      +21.03  "  (brasses).     .     . 
34.92       "       +  65.08  "  (small  bells)     . 
15.17       "      +84.83  "  (speculum  metal) 
Tin                                                      .    . 

8.561 
8.462 

8-459 
8.728 
8.056 
7447 

7  2QI 

36060 
34048 
39648 
30464 

3T36 
6944 
c6oo 

Zinc      .                             .          .... 

686l 

7SOO 

Copper  cast  

8.712 

24138 

8.878 

77OOO 

Copper  wire                •              •     . 

60000 

I7OOOOOO 

Gold  cast  

IQ  2C8 

2OOOO 

IO.476 

40000 

22.069 

56000 

Lead  cast      

1  1  7  C2 

1800 

the  oak,  pine,  etc.,  we  shall  find  a  series  of  concentric  layers  ; 
these  layers  being  called  annual  rings,  because  one  is  generally 
deposited  every  year. 

Radiating  from  the  heart  outwards  will  be  found  a  series  of 
radial  layers,  these  being  known  as  the  medullary  rays. 

Of  the  annual  rings,  the  outer  ones  are  softer  and  lighter  in 
color  than  the  inner  ones ;  the  former  forming  the  sap-wood,  and 
the  latter  the  heart-wood.  When  the  log  dries,  and  thus  tends 
to  contract,  it  will  be  found  that  scarcely  any  contraction  takes 
place  in  the  medullary  rays ;  but  it  must  take  place  along 
the  line  of  least  resistance,  viz.,  along  the  annual  rings,  thus 
causing  radiating  cracks,  and  drawing  the  rays  nearer  together 
on  the  side  away  from  the  crack.  This  action  is  exhibited  in 
Fig.  241,  where  a  log  is  shown  with  two  saw-cuts  at  right 
angles  to  each  other ;  when  this  log  bf  ^mes  dry,  the  four 


STRENG7*H   OF    TIMBER. 


645 


FIG.  241. 


right  angles  all  becoming  acute 
through  the  shrinkage  of  the 
rings. 

If  the  log  be  cut  into  planks  by 
parallel  saw-cuts,  the  planks  will, 
after  drying,  assume  the  forms 
shown  in  Fig.  242,  as  is  pointed 
out  in  Anderson's  "  Strength  of 
Materials,"  from  which  these  two 
cuts  are  taken. 

This  internal  construction  of  a 
plank  has  an  important  influence 
upon  the  side  which  should  be  uppermost  when  it  is  used  for 

flooring  ;  for,  if  the  heart  side  is  up- 
permost, there  will  be  a  liability  to 
having  layers  peel  off  as  the  wood 
dries  :  indeed,  boards  for  flooring 
should  be  so  cut  as  to  have  the  an- 
nual rings  at  right  angles  to  the 
side  of  the  plank.  Before  discuss- 
ing any  other  considerations  which 
affect  the  adaptability  of  timber  to 
use  in  construction,  we  will  con- 
sider the  question  of  its  strength. 
§  237.  Strength  of  Timber.  —  In  this  regard  we  must 
observe,  that,  whereas  the  strength  and  elasticity  and  other 
properties  of  iron  and  steel  vary  greatly  with  its  chemical  com- 
position and  the  treatment  it  has  received  during  its  manufac- 
ture, the  strength,  etc.,  of  timber  is  much  more  variable,  being 
seriously  affected  by  the  soil,  climate,  and  other  accidents  of  its 
growth,  its  seasoning,  and  other  circumstances  ;  and  that  over 
many  of  these  things  we  have  no  control :  hence  we  must  not 
expect  to  find  that  all  timber  that  goes  by  one  name  has  the 
same  strength,  and  we  shall  find  a  much  greater  variation  and 


FIG.  242. 


646  APPLIED    MECHANICS. 

irregularity  in  timber  than  in  iron.  The  experiments  that  have 
been  made  on  strength  and  elasticity  of  timber  may  be  divided 
into  the  following  classes  :  — 

i°.  Those  of  the  older  experimenters,  except  those  made 
on  full-size  columns  by  P.  S.  Girard,  and  published  in  1798. 
A  fair  representation  of  the  results  obtained  by  them,  all  of 
which  were  deduced  from  experiments  on  small  pieces,  is  to 
be  found  in  the  tables  given  in  Professor  Rankine's  books, 
"  Applied  Mechanics,"  "  Civil  Engineering,"  and  "  Machinery 
and  Millwork." 

2°.  Tests  made  by  modern  experimenters  on  small  pieces. 
Such  tests  have  been  made  by  — 

(a)  Trautvvine  :  Engineers'  Pocket-Book. 
(b}   Hatfield  :  Transverse  Strains. 
(c)   Laslett :  Timber  and  Timber  Trees. 
(</)  Thurston  :  Materials  of  Construction. 

(e)  A  series  of  tests  on  small  samples  of  a  great  variety  of  American 
woods,  made   for   the  Census  Department,  and   recorded  in 
Executive  Document  No.  5,  48th  Congress,  ist  session. 
Timber  Physics,  Division  of  Forestry,  U.  S.  Department  of  Agri- 
culture.   For  a  fairly  complete  bibliography  of  tests  of  tim- 
ber see  a  paper  by  G.  Lanza,  Trans.  Am.  Soc.  C.  E.,  1905. 

3°.  Tests  made  by  Capt.  T.  J.  Rodman,  U.S.A.,  the  results 
of  which  are  given  in  the  "  Ordnance  Manual." 

4°.  All  tests  that  have  been  made  on  full-size  pieces. 

In  regard  to  tests  on  small  pieces,  such  as  have  commonly 
been  used  for  testing,  it  is  to  be  observed,  that,  while  a  great 
deal  of  interesting  information  may  be  derived  from  such  tests 
as  to  some  of  the  properties  of  the  timber  tested,  nevertheless, 
such  specimens  do  not  furnish  us  with  results  which  it  is  safe 
to  use  in  practical  cases  where  full-size  pieces  are  used.  Inas- 
much as  these  small  pieces  are  necessarily  much  more  perfect 
(otherwise  they  would  not  be  considered  fit  for  testing),  having 
less  defects,  such  as  knots,  shakes,  etc.,  than  the  full-size  pieces. 


Sl^RENGTff  OF   TIMBER. 


647 


they  have  also  a  far  greater  homogeneity.  They  also  season 
much  more  quickly  and  uniformly  than  full-size  pieces.  In 
making  this  statement,  I  am  only  urging  the  importance  of 
adopting  in  this  experimental  work  the  same  principle  that  the 
physicist  recognizes  in  all  his  work ;  viz.,  that  he  must  not 
apply  the  results  to  cases  where  the  conditions  are  essentially 
different  from  those  he  has  tested. 

Moreover,  it  will  be  seen  in  what  follows,  that,  whenever 
full-size  pieces  have  been  tested,  they  have  fallen  far  short  of 
the  strength  that  has  been  attributed  to  them  when  the  basis 
in  computing  their  strength  has  been  tests  on  small  pieces  ; 
and,  moreover,  the  irregularities  do  not  bear  the  same  propor- 
tion in  all  cases,  but  need  to  be  taken  account  of. 

The  results  of  the  first  class  of  experiments  named  in  the 
following  table  are  taken  from  Rankine's  "  Applied  Mechanics ;" 
and,  inasmuch  as  the  table  contains  also  the  strengths  of  some 
other  organic  fibres,  it  will  be  inserted  in  full.  The  student 
may  compare  these  constants  with  those  that  will  be  given 
later. 


Kind  of  Material. 

Tenacity 
or  Resist- 
ance to 
Tearing. 

Modulus  of 
Tensile 
Elasticity. 

Resist- 
ance to 
Crush- 
ing. 

Modulus 
of 
Rupture. 

Resist- 
ance to 
Shearing 
along 
Grain. 

Modulus 
oi 
Shearing 
Elasticity 
along  the 
Grain. 

Ash        

I7OOO 

1600000 

QOOO 

(  12000 

I4OO 

76000 

Bamboo                    .     . 

(  I4OOO 

Beech    

II5OO 

I-KOOOO 

0160 

(     9000 

1     _ 

Birch               .     . 

1  5OOO 

1  64  sooo 

6400 

(  I2OOO 
I  I7OO 

' 

Blue  cum 

8800 

(  IbOOO 

1    _ 

Box  

2OOOO 

10300 

(  2OOOO 

f 

Bullet-tree      .... 
Cedar  of  Lebanon  .     . 

II4OO 

486000 

14000 
5860 

(15900 
|   16000 
7400 

': 

- 

648 


APPLIED   MECHANICS. 


Kind  of  Material. 

Tenacity 
or  Resist- 
ance to 
Tearing. 

Modulus  of 
Tensile 
Elasticity. 

Resist- 
ance to 
Crush- 
ing. 

Modulus 
of 
Rupture. 

Resist- 
ance to 
Shearing 
along 
Grain. 

of         1 
Shearing    I 
Elasticity 
along  the 
Grain. 

(  IOOOO 

) 

Chestnut    

to 

/•  1140000 

- 

10660 

- 

- 

(  13000 

) 

Cowrie  

- 

- 

- 

IIOOO 

- 

- 

Ebony              .... 

_ 

_ 

I9OOO 

•77000 

(     700000 

)• 

(          6000 

) 

Elm                 .     . 

14000 

5        to 

)     to 

<           IU 

76000 

\               LV-' 

(  1840000 

)    °30C 

(     9700 

)M 

Fir,  Red  pine      .     .     . 

(  12000 

to 
f  14000 

1460000 
to 
1900000 

5375 
to 
6200 

r    7100 
^    9540 

500 
800 

62OOO 
II6OOO 

"     Yellow  pine  (Am.) 

- 

- 

5400 

- 

- 

- 

"     Spruce    .... 

12400 

(  1400000 
to 
(  1800000 

|  - 

j    9900 
)  12300 

|     600 

- 

"     Larch     .... 

(  9000 

to 

(  IOOOO 

900000 
to 
1360000 

|  5570 

1  IOOOO 

970 
I7OO 

\    - 

Hoxen  yarn    .... 

25000 

- 

- 

- 

- 

- 

Hazel 

16000 

(   I2OOO 

) 

Hempen  rope     .     .     . 

to 

( 

_ 

_ 

_ 

_ 

(  16000 

) 

Ox-hide,  undressed 

6300 

_ 

- 

- 

- 

- 

Hornbeam      .... 

20000 

_ 

- 

- 

- 

- 

Lancewood     .... 

23400 

- 

- 

- 

- 

- 

Ox-leather      .... 

42OO 

24300- 

- 

- 

- 

- 

Lignum-vitae  .... 

IISOO 

- 

9900 

I2OOO 

- 

- 

Locust        ...          . 

l6oOO 

_ 

Mahogany 

(     8000 

<     to 

\  i  2  e  CQOO 

8200 

j  7600 

)      •" 

(  21800 

) 

-> 

Maple 

10600 

Oak,  British  .... 

- 

IOOOO 

(  IOOOO 

]  13600 

"     Dantzic      .     .     . 

- 

- 

7700 

8700 

"     European  .     .     . 

(  IOOOO 

to 

(  19800 

f  1200000 

C  1750000 

}  - 

- 

•  2300 

82OOO 

"     American  red 

10250 

2I5OOOO 

6000 

10600 

STRENGTH  OF    TIMBER. 


649 


Resist- 

Modulus 

Kind  of  Material. 

Tenacity 
or  Resist- 
ance to 
Tearing. 

Modulus  of 
Tensile 
Elasticity. 

Resist- 
ance to 
Crush- 
ing. 

Modulus 
of 
Rupture. 

ance  to 
Shearing 
along 
Grain. 

of 
Shearing 
Elasticity 
along  the 

Grain. 

Silk  fibre   

52OOO 

I3OOOOO 

I  7OOO 

I  O4OOOO 

0600 

Teak,  Indian       .     .     . 

I5OOO 

24OOOOO 

I2OOO 

(  I2OOO 
\  IQOOO 

1- 

- 

African     .     .     . 

2IOOO 

2300000 

- 

14980 

- 

- 

Whalebone     .... 

7700 

- 

- 

- 

- 

- 

Willow            .... 

6600 

Yew  .          ..... 

8000 

in  regard  to  the  tests  of  the  second  class,  a  few  comments 
are  in  order  :  — 

i°.  These  experiments,  like  those  of  the  first  class,  were  all 
made  upon  small  pieces  ;  and  the  results  are  correspondingly 
high. 

The  usual  size  of  the  specimens  for  crushing  being  one  or 
two  square  inches  in  section,  and  of  those  for  transverse 
strength  being  about  two  inches  square  in  section  and  four  or 
five  feet  span,  those  for  tension  had  even  a  much  smaller  sec- 
tion than  those  for  compression  ;  as  it  is  necessary,  in  order  to 
hold  the  wood  in  the  machine,  to  give  it  very  large  shoulders. 

The  only  exception  to  this  is  the  tests  of  Sir  Thomas  Las- 
lett,  an  account  of  which  is  given  in  his  "Timber  and  Timber 
Trees,"  and  also  in  D.  K.  Clark's  "  Rules  and  Tables."  In  these 
tests  he  gives  very  much  lower  tensile  strengths  than  those 
given  above ;  and  he  states  that  his  specimens  were  three  inches 
square,  but  does  not  say  how  he  managed  to  hold  them  in  such 
a  way  as  to  subject  them  to  a  direct  tensile  stress.  His  results 
for  crushing  and  transverse  strength  are  about  as  great  as 


650  APfLIED    MECHANICS. 


those  given  in  Rankine's  tables,  and  as  were  obtained  by  the 
other  experimenters  on  small  pieces,  as  his  specimens  were  of 
about  the  same  dimensions  as  those  used  by  the  others.  The 
figures  obtained  by  these  experimenters  will  only  be  given  inci- 
dentally, as  — 

(a)  They  are  very  similar  to  those  given  in  Rankine's  table. 

(b)  They  are  not  suitable  for  practical   use  on   the  large 
scale. 

(c)  While  they  have  been  used,  it  has  only  been  done  by 
employing  a  very  large  factor  of  safety  for  timber. 

The  series  of  tests  made  for  the  Census  Department,  and 
recorded  in  Executive  Document  No.  5,  48th  Congress,  first 
session,  form  a  very  interesting  series  of  experiments  upon 
small  specimens  of  an  exceedingly  large  number  of  .America0, 
woods.  In  order  to  have  working  figures,  we  should  need  to 
test  large  pieces  of  the  same  ;  as  the  proportion  between  the 
strengths  of  the  different  kinds  would  be  liable  to  be  different 
in  the  latter  case. 

The  work  done  by  the  Division  of  Forestry  of  the  U.  S. 
Dept.  of  Agriculture  before  1898  was  mostly  of  this  class,  but 
little  having  been  done  with  full-size  pieces,  and  that  with 
imperfect  apparatus. 

The  only  record  of  Rodman's  experiments  available  is  a 
table  of  results  in  the  "  Ordnance  Manual."  These  are  lower, 
as  a  rule,  than  those  obtained  by  the  experimenters  of  the  first 
or  second  class.  This  is  to  be  accounted  for  by  the  fact  that, 
while  he  did  not  experiment  on  full-size  pieces,  he  used  much 
larger  pieces  than  those  heretofore  employed  ;  his  specimens 
for  transverse  strength,  many  of  which  are  still  stored  at  the 
Watertown  Arsenal,  being  5f  inches  deep,  2$  inches  thick,  and 
5  feet  span. 

The  fourth  class  of  tests  are  those  which  furnish  reliable 
data  for  use  in  construction  ;  'and  we  will  proceed  to  a  consid- 
eration of  these,  taking  up  (1°)  tension,  (2°)  compression,  (3°) 
transverse  strength,  and  (4°)  shearing  along  the  grain. 


TENSION.  65  I 


TENSION. 

In  all  cases  where  the  attempt  has  been  made  to  experiment 
upon  the  tensile  strength  of  timber,  a  great  deal  of  difficulty 
has-been  encountered  in  regard  to  the  manner  of  holding  the 
specimens.  In  all  cases  it  has  been  found  necessary  to  pro- 
vide them  with  shoulders,  each  shoulder  being  five  or  six  times 
as  long  as  the  part  of  the  specimen  to  be  tested,  and  to  bring 
upon  these  shoulders  a  powerful  lateral  pressure,  to  prevent 
the  specimen  from  giving  way  by  shearing  along  the  grain,  and 
pulling  out  from  the  shoulder,  instead  of  tearing  apart. 

The  specimens  tested  have  generally  had  a  sectional  area 
less  than  one  square  inch,  and  it  seems  almost  impossible  to 
provide  the  means  of  holding  larger  specimens.  This  being 
the  case,  it  is  plain,  that,  whenever  timber  is  used  as  a  tie-bar 
in  construction  (except  in  exceedingly  rare  and  out-of-the- 
way  cases),  it  will  give  way  by  some  means  other  than  direct 
tension  ;  i.e.,  either  by  the  pulling-out  of  the  bolts  or  fastenings, 
and  the  consequent  shearing  of  the  timber,  or  else  by  bending 
if  there  is  a  transverse  stress  upon  the  piece  ;  and,  this  being 
the  case,  these  other  resistances  should  be  computed,  instead 
of  the  direct  tension.  Hence,  while  the  direct  tensile  strength 
of  timber  may  be  an  interesting  subject  of  experiment,  it  can 
serve  hardly  any  purpose  in  construction  ;  and  the  conclusion 
follows,  that  the  resistances  of  timber  to  breaking  we  may 
expect  to  meet  in  practice  are  its  crushing,  transverse,  and 
shearing  strength.  Indeed,  the  use  of  timber  for  a  tie-bar 
should  be  avoided  whenever  it  is  possible  to  do  so  ;  and,  when 
it  is  used,  the  calculations  for  its  strength  should  be  based 
upon  the  pulling-out  of  the  fastenings,  the  shearing  or  splitting 
of  the  wood,  etc.,  and  not  on  the  tensile  resistance  of  the  solid 
piece. 

Moreover,  when  a  wooden  tie-bar  is  subjected  not  merely 
to  direct  tension,  but  also  to  a  bending-moment,  whether  the 
latter  is  caused  by  a  transverse  load,  or  by  an  eccentric  pull,  as 
it  generally  is  in  the  case  of  timber  joints,  we  must  compute 


652  APPLIED    MECHANICS. 

the  greatest  tension  per  square  inch  at  the  outside  fibre  due  to 
the  bending,  and  to  that  add  the  direct  tension  per  square  inch: 
and  this  sum  must  be  less  than  the  modulus  of  rupture  if  the 
piece  is  not  to  give  way;  i.e.,  the  modulus  of  rupture  and  not 
the  ultimate  tensile  strength  per  square  inch  must  be  our  criterion 
of  breaking  in  such  a  case,  the  working- strength  per  square  inch 
being  the  modulus  of  rupture  divided  by  a  suitable  factor  of  safety. 

COMPRESSIVE    STRENGTH. 

Tests  of  the  compressive  strength  of  full-size  wooden  columns, 
with  the  exception  of  one  set  of  tests,  date  from  about  1880. 

TESTS   OF   FULL-SIZE   COLUMNS. 

The  following  are  references  to  tests  of  full-size  timber  columns: 

i°.  Trautwine,  in  his  "Handbook,"  speaks  of  some  tests  of 
wooden  pillars  20  feet  long  and  13  inches  square,  made  by  David 
Kirkaldy,  which,  as  he  says,  gave  results  agreeing  with  Mr.  C. 
Shaler  Smith's  rule. 

2°.  A  series  of  tests  made  at  the  Watertown  Arsenal  for  the 
Boston  Manufacturers'  Mutual  Fire  Insurance  Company,  under 
the  direction  of  the  author. 

3°.  Eleven  tests  of  old  spruce  pillars  made  at  the  Watertown 
Arsenal,  for  the  Jackson  Company,  under  the  direction  of  Mr. 
J.  R.  Freeman,  and  reported  in  the  Journal  of  the  Assoc.  Eng. 
Societies  for  November,  1889. 

4°.  The  tests  that  have  been  made  at  the  Watertown  Arsenal 
on  the  government  testing-machine. 

5°.  Tests  made  in  the  Laboratory  of  Applied  Mechanics  of 
the  Massachusetts  Institute  of  Technology. 

6°.  A  series  of  tests  of  full-size  columns  of  oak  and  fir,  made 
by  P.  S.  Girard  in  1798. 

In  regard  to  the  first,  no  details  or  results  are  given:  hence 
nothing  will  be  said  about  them. 

In  regard  to  the  second,  a  summary  only  will  be  presented 
here. 


TESTS  OF   YELLOW-PINE  POSTS  AND  BLOCKS.         653 


cq  S  ^  «5  <n  co   eo 

"  -4->  'O  "D  t3  'u  "o 

c,acc;i*'"ccGcccro£.Crt4>  c  c  c  c 

*j^>**jt«c*j^+j*j*j*JDx:73-w4)c:  *j  **  -M  .w 

rt  rt  CTJ  rt  g  ."tii  C™c^  ^  ^  rtc^ 

7-1  ^    'aE  (5  2  E  E  E  E 


•^• 

VO 

N 


i-ito>-irON 

N    N    M    to  to 


m 


10  —  r^.  -^-     1 


00   fOOO   O  vo  00  m  -^-  rt  O 

00  VO   O   N  «    OVO   T}-VO   ON 


i    i 


'_) 
CQ 

Q 

?  s 

w    * 

H    W 

S  5 
fc  3 

u  fe 

?! 
§ 

2j 

>,   H 

b 

O 


^6- 

Is^ 


>-i\O 
VO  O 


M 

to 


T^- 

O 

O 


fl43 

It 

2  8 


v    • 


QQOO  ^no*MOfO  t^>vO 

ON^"^O  r^    WI^ONM  co\O 

tow    N    C)  CO      M    -^-  VOOO    C^J    tv. 

VO  *0  TI-  to  00     VO  -<d-oo  l^vO  -<t 


«      VO 

q    *j 


O       VO  O 

«>    ^p 


&.&. 


j-S 


ioq  osrx.  ^5  j^ 

d  dod  r^.    i      i    i    i    i    i    i     o  3     i 

e^ 


"§8 

w- 


,  to       co  O     O    oo        c^  O        r^  O ' 
O  \f   *+   ON     O      ON  \s   Q  w  ^s   Q  t^x 

d      dt^oood      cKd      dr^. 


.100^00      O      OOfOOfOQ        "^       Q      •<*•          O 
CWNWCJ     q     cjr^wqroo        n        WNO          9 

"Nddddd^«dddd      d      M«        d 


wTj-N 

1-1  vo      rf 


o      rf     Tj-oo 
i-irONw 


oo       ro    to 

-^  N  HH 


II 


000    0^^^£         000 

P-i        PH     PH 


8    ^^ 

PQ      PQ  P3 


;as  ;si     -jas  pz     -}3s  pz 


654 


APPLIED   MECHANICS. 


H 


S   i 


t4    a 
<    t 


,2      JS       *4      ^ 

C         G         C         C 

|| 

^'f  ^    -s 

*>  c       c         c 

•           •           • 

M           tft           Oi 

c       c       c 

E    E    E    E 

en   §        •!-•           ->-> 
0)   »-.         rt           rt 

h       E      E 

E    E    E 

fill 

N          OO          00           -i 
N           ON          CO        OO 
W           CO          -^        OO 
N            CO          O           •^- 

M        vo         m        r^ 

1              00               1 

^         ^         J? 

co       ON      vO 

to          ^-          N 
VO         O         O 

w           w            CO 

"rrt                          4>         . 

pc75       ^  *""* 

vo       oo       vo       oo 

O         OO           ^0         CO 

CO        co        co        co 

00                ON              O 
CO             *"*               tO 

M               CO             ^J- 

CO        CO        O 
CO       co       co 

1 

CO 

ft        » 

III! 

O           to         vO 

TJ-      vO        to 

• 

1  "s-  s  s 

3,  ^    ^    3 

•^-        oo 
*>        Q         •* 

VO               to             CO 

CO         to         CO 
N           to         ON 
00           N         VO 

. 

c£  <    ""  -S 

ill 

3*8.5 

vO        vO        vO        VO 

?  •§  ' 

1              1              1 

• 

Q  $:s 

d       d       o\     06 

1          1          1 

1              1              1 

. 

*      1  8 

jb|1 

o\     06       tx     vo 

O           i^          ON 
0           r^          0 

oo      oo       co 
ON      06        t^. 

III 

||.l 

.    r^        0         0         O 

e    I-H        oj        o        M 

"dodo 

8      8      8 

odd 

vo        O       oo 
O         O        OO 

d      d      « 

N           N           *H 

• 

fl 

to        >O        ON        *O 

ON       M       oo         >-> 

£n        8         £. 

ON          w            CO 

V 

bJO 

A  a1 

••a  £   6 
«!* 

M              N              f)            Tf 

C/5            C/3            C/5            C/5 
O          O          O          O 
PH        (X,        Ps        PH 

*J                    4J                     CJ 
Oi                 fO                 Q 

(S      c2      5 

CJ            CO          '^f 
CJ           O           CJ 

PQ      P3      PQ 

> 

•49SJSI 

•}9S  pz 

•jas  pj 

TESTS  OF  OLD   AND   SEASONED    WHITE-OAK  POSTS.    655 


8 

•C-CHC      "§ 

rt   rt-0   rt         rt 

D    CJ    C 
,0.0     <U     O-  OH 


ng 
ng 
ari 
d 


rt  rt  rt  ** 

•s 


en 
en 
ev 
ca 


4^   tl   G    G 


>   >   > 


<u  5  <u  U73  <u  v  <u  3 

<u      <u      ooo>oqr°ooo^ 

~    &  rrt:  ^r.s  «rrr  S 

rt  r*    n    i-*    rt    *-"    X  ^C    ,— i    «~i    <•-*  -4-» 

o  o 


•qouj  aiunbg 
jad'-sqjui'Ajpp 
-sei3  jo  snjnpop^ 


Tf  rf 

N  *O 

CXD  •^- 

r^  N 

rf  ro  >^>  *o 

00   «  VO  O 

1-1   N    1-1  N 


I    I    i    i    I 


•qouj  aaenbg 


vO  O  *2 


ro       O     lo  O  "^J"  "^"  r^4    oo      *"*  ro  »^  vO  t^. 

N        LovOoOOQQvO      roco   roco  O   r}- 
^    °5.    ^^^^  •=• 


•saqoui  arenbs 
ui  ' 


asa 


O  O  oo  ovovoor^oo  TJ-  r^vo  ro  ro 

gS  ii  q  rj-r^Ntv-HHvq  o^ro  LOCO  q 

N  10  vd  TJ-rot-^-^-Ovw  ^  ^"VO  ^ovd 

N  M  w  oooocooor^co  VOVOVOVON 


•ssqoui  ui 

'3JCT)  JO  J3J3UIEIQ 


lOQO     !O       »o       N        O      OO^OOO      O     LoOOQn 

qvONq    ON      qv      qv      q     q\q>ON  q\cq    r^.    ON  ON  q\  ONVO 


•ssqom 
ui  'pug;  3§JBq 

3qj    JO     J3J3UIEIQ 


Tj-   U-)O          LO  CO 

cooo  r^t^oq       f>     ^.      \    \    t    \    \       i       ill 

3       VO       VO       VO 


CO 

°°. 


•ssqoui 
ui  *pug   ||Btug 

3qi    JO    J313UIEIQ 


- 

oq  oq  oq     q 

\j~l  If)  IT)     VO 


S^    2 


ro  N   Tf  rocs 


•o      vo     OOOOO     O 


•ssqoui  pus 


W    W    O       O  O  O          *-O      O    O    O    O    ^h      CO      O  OO 

qvON^-v5      oq       «      vq    i>:T9^9vO.     ^9 
dd«6      d      d     cod.  oSdd'-Iwodco 


«J   C<    M 


ro  ro  ro  ro  ro  w 


•sqi    ui 


O     CO 

ft      CO 


SuiqsmSuiisiQ 


CT\ONON  O   ON   ON   ON  •<  OG  O  Q  UI  U_ 


656 


APPLIED   MECHANICS. 


In  all  the  experiments  enumerated  in  the  tables  given 
above,  the  columns  gave  way  by  direct  crushing,  and  hence 
the  strength  of  columns  of  these  ratios  of  length  to  diameter 
can  properly  be  found  by  multiplying  the  crushing-strength  per 
square  inch  of  the  wood  by  the  area  of  the  section  in  square 
inches. 

This  conclusion  is  deduced  from  the  fact  that  the  deflections 
were  measured  in  every  case,  and  found  to  be  so  small  as  not  to 
exert  any  appreciable  effect. 

In  regard  to  other  tests  of  this  same  set,  there  were  eight 
tests  made,  in  addition  to  those  already  enumerated ;  and  in 
five  the  loads  were  off  centre.  A  summary  of  the  results  is 
appended,  together  with  a  comparison  of  their  actual  strength 
with  that  which  would  be  computed  on  the  basis  of  4400  per 
square  inch  for  yellow  pine,  and  3000  for  oak.  The  first  three 
tests  were  made  on  yellow-pine  columns,  and  the  last  two  on 
oak. 


Weight, 
in  Ibs. 

Length,  in 
feet  and 
inches. 

Diameter 
of 
Column. 

Diam- 
eter of 
Core. 

Sectional 
Area,  in 
square 
inches. 

Eccen- 
tricity, 
in 
inches. 

Ultimate 
Strength. 

Computed 
Ultimate 
Strength. 

ft.          in. 

2,  2d  series 

320 

II     11.27 

9.92 

i-53 

7545 

2-33 

265000 

331980 

5,  3d  series 

298 

12     6.8 

(    8.30) 

\       X    \ 
I    7-60  ) 

- 

63-1 

2.07 

240000 

277640 

i,  3d  series 

386 

12       9.3 

(   8.75) 

\      X  ( 

(   8.92) 

- 

76.04 

2.25 

280000 

334576 

i,  2d  series 

451 

II      II.4 

10.95 

i.  80 

92.16 

2-75 

170000 

276480 

3,  2d  series 

236 

II      II.  2 

8.2 

i-55 

50.92 

1.91 

100000 

152760 

These  results  exhibit  a  great  falling-off.  of  strength  due  to 
the  eccentricity  of  the  load  ;  and  emphasizes  the  importance 
of  taking  into  account  eccentric  loading  in  our  calculations  in 
a  manner  similar  to  that  already  mentioned  on  pages  370,  371, 
and  448. 


STRENGTH  OF   TIMBER.  6$/ 

The  remaining  experiments  were  :  (i°)  Two  tests  of  white- 
wood  columns,  average  strength  3000  pounds  per  sq.  in.,  and 
very  brittle.  (2°)  One  yellow-pine  square  column  (sectional  area 
68.8  sq.  in.,  length  12'  6".85)  with  one  end  resting  against  a 
thick  yellow-pine  bolster. 

The  maximum  load  was  120,000  Ibs.  =  1744  Ibs.  per  sq. 
in.,  the  post  beginning  to  split  due  to  eccentricity  of  bearing 
caused  by  uneven  yielding  of  the  bolster.  The  bolster  was 
then  removed,  the  post  cut  off  i^  in.  at  the  end  and  tested 
without  the  bolster.  Ultimate  strength  375,000  Ibs.  =  5451 
Ibs.  per  sq.  in. 

The  table  of  results  of  the  tests  on  old  and  seasoned  oak 
columns  were  made  upon  columns  that  had  been  in  use  for  a 
number  of  years  in  different  mills,  from  which  they  were  re- 
moved, and  replaced  by  new  ones.  Ten  of  them  had  been  in 
use  about  twenty-five  years,  and  the  remainder  for  shorter 
periods.  An  inspection  of  this  table  will,  I  think,  convince  the 
reader  that  it  would  not  be  safe  to  calculate  upon  a  higher 
breaking-strength  per  square  inch  in  these  than  in  the  green 
ones. 

TESTS    FOR    THE   JACKSON    COMPANY. 

Eleven  tests  of  old  spruce  pillars,  which  had  been  in  use 
in  a  cotton-mill  of  the  Jackson  Company,  were  tested  on  the 
government  machine  at  Watertown,  under  the  direction  of 
Mr.  J.  R.  Freeman.  The  manner  of  making  them  was  as  fol- 
lows: 

In  the  first  two  the  ends  were  brought  to  an  even  bear- 
ing. 

In  the  third  the  ends  came  to  an  even  bearing  under  a  load 
of  60000  pounds. 

In  the  fourth,  fifth,  ninth,  tenth,  and  eleventh,  the  cap, 
and  also  the  base-plate,  were  planed  off  on  the  back  to  a  slope 
of  i  in  24,  and  placed  with  their  inclinations  opposite. 

In  the  eighth  they  had  their  inclinations  the  same  way  one 
as  the  other. 


658 


APPLIED  MECHANICS. 


In  the  sixth  and  seventh  the  base-plate  was  not  used,  the 
larger  end  of  the  post  having  a  full  bearing  on  the  platform  of 
the  machine. 

The  results  are  given  in  the  following  table : 


Diameter 

Diameter 

Ultimate 

Length 
in  Feet  and 
Inches. 

at 
Small  End, 
Inches. 

at 
Large  End, 
Inches. 

Area  at 
Small  End, 
Sq.  In. 

Ultimate 
Strength, 
Lbs. 

Strength  per 
Sq.  In., 
Lbs. 

ft.        in. 

I 

10     4-75 

5.82 

7.78 

31.87 

142000 

4088 

2 

10     4. 

5-85 

7-49 

27.15 

192800 

6225 

3 

10     5-75 

5.85 

7-74 

32.17 

166100 

4900 

4 

10     5-5 

5-70 

7-77 

31.67 

108200 

5 

10     5.1 

5-70 

7.70 

30.78 

105000 

6 

TO      5.2 

5.80 

7.61 

30.39 

168000 

7 

9     7 

5-74 

7-81 

32.88 

194100 

8 

10     5.4 

5-85 

7.90 

34-21 

T55000 

9 

10    4.9 

5-82 

7-77 

40.72 

96100 

10 

10     5-13 

5-73 

7.78 

125000 

ii          10    4.38 

5-74 

7.81 

60000 

All  but  the  first  three  of  the  tests  were  made  with  inclined 
bearings  of  one  kind  or  another,  hence  the  ultimate  strength 
per  square  inch  is  only  given  here  for  the  first  three ;  which,  as 
Mr.  Freeman  says,  were  of  "  well-seasoned  spruce,  of  excellent 
quality."  Hence  the  average  crushing-strength  of  spruce  is 
doubtless  considerably  lower  than  the  average  of  these  three. 


TESTS    MADE    ON    THE    GOVERNMENT    MACHINE. 

In  Executive  Document  12,  47th  Congress,  first  session, 
will  be  found  a  series  of  tests  of  white  and  yellow  pine  posts 
made  at  the  Watertown  Arsenal ;  and  these  tests  probably  fur- 
nish us  the  best  information  that  we  possess  in  regard  to  the 
strength  of  wooden  columns. 

The  summary  of  results  is  appended  :  — 


COMPRESSION  OF   WHITE-PINE   POSTS. 


659 


If 


1  i 

' 


S       -       5 


Q 


^ 

'"".   d    c>    c> 

crrorOfO 


OOOOU">L'-> 

i-Hr^r^cxDMw 
d    CNONcKd    c5 

fONNNCOrO 


Ooooo     M     r^    to   ON   w    to  vo     O     N     t^oo    ro   to  oo    to  vo    r~> 

•  vc*'^"'^"L9'^"T?"T?"'^"'^"T?'ijr'vc>T?"'^"fr5rPf/>r7>^  i 


Ooooo     O     r^OOOOO     tovo     Ooo     r--oo     rotoo     moo     a 

^«^^^iOTj-tO-^--^--^-Tj-to-^-Ti--^-fOfO^-fON      Cl 


^r 


o    o    cJ 

O     O     O 


s-  m^ 

odd 


d    d 
8   § 


•<        O 


1       1       1       1       1       1       1       1       1       1       1       1       1 


O    voo     tovoo 


o  to  q    q    «?  o    q 


d<~  to 


S    ox  ax 


O\OvO     r^.OO     ro-^-toi-i     N     ro-^- 


660 


APPLIED  MECHANICS. 


o  5    5    5 
«-  »o 

2  •  «•  5  " 

sr::  $^2     * 


s  < 


'J5»    .  ** 

S       O    'S  O    3      3      - 

W)    5  hO 

'S  "53  'S 

S      S      3  S      S      S      3 


S55353523 


£s 


CO   f>  v£>     LO 
ro   i^*    ^     ON 


M      M      M      M      M 


r^    N   >S     O    »H 
M     M     i-     ro    M 


vo 


ION     N 


OO 
N 


vooo     r~^oo    « 
fO\O     NOMD 


•5  S 


il 

M       M 


M       M 


Q 

vo 


OQQQOOOOOQQQO 

TJ-\O^O    ONt^^vo    fofovqvo    q    co 
M    M    M    r^tv.r^t^r>.t^t^t^^O    t^ 


M     w     N     M     M     roforovo    t^vOMD    r^r^t^t-^M 


3, 


M     M     M     rorotOt-^r^ 


—   dvdvdvd    d    d    d 


»oo     O     r^O     f 

rO'^J-fOO|-|«i-ip-i«'-it-i 

vdvdvdododod    d    d    d    •^••^• 


MMNMMNNNN 


O     O     O 


a,  0 

W— 
•§§, 


|       1       1       1       1 


qq 

mO 


qo 

rood 


qqoqooooiooooqo»oo 

r^tor^ONfOt^rJ-fodvdvO 

M      N      N\O      r^'OONONWVOvOV 


COMPRESSION  OF   WHITE-PINE   POSTS. 


66 1 


-d 
1 


. 


11 
fl 


I|I|.S 

-S   §  .S   g       jg 


s 


rovOvO     w     r^N 


OOOOOQOOOOOOOQ 

^NiooqqqNWLOi-cwNqwvq 
r^t^r^^O    r^.'Ooooooo     O     O    O    roro 


oo    "nM 
.    iot^vO 


w     OMD    -*r^oo     O     O     -<trtQ 
^o    lOfOroro^ovovOvOvOvO 


loinvnvdvdvdooodod 


—     —     1-1 


&w   4 

l$*i 


^qqqqqqqqqqqqqqqqqqq 
O    ^foo    ^O^ON     i^Q     r^r^^O    t^t^.vO'-ovO    r-^r^tx 
M    r^r^oo     O\"^"^^-N     ONOO    OvOvO    »o 
N     N     N     N     N     rofOrO-^-fOrorON     M     fJ 


i^-OO     N     fOTj- 
rofOO     O     O 


O     OO     O 
tv.t^rxi>. 


662 


APPLIED   MECHANICS. 


COMPRESSION    OF    WHITE    PINE.  —  SINGLE  STICKS  AND  BUILT  POSTS. 
In   the   multiple  ones,   dimensions  of   each   stick   are   given. 


No. 
of 
Test. 

Weight. 

*5 

«  1 

Si* 

22& 
£O.S 
<j    « 

Dimensions  of  Post. 

Sectional" 
Area. 

•S|R 

.2     ^ 
1"  8- 

i*r* 

§  M  II  in 

Ultimate  Strength. 

i 
j 

1 

•s 

a 
Q 

•a 

3 

S 

< 

^ 
fi 

Ibs. 

in. 

in. 

in. 

sq.  in. 

in. 

Ibs. 

664 

i53 

ii 

i77'5o 

4.48 

11.65 

52.2 

0-0545 

IIOOOO 

2107 

665 

i43 

IO 

180.00 

4.48 

11.64 

52-1 

O.IOIO 

81500 

^64 

666 

163 

5 

179.97 

4-47 

11.63 

52.0 

0.0895 

70000 

1346 

667 

228 

13 

180.00 

5-40 

11.30 

61.0 

0.0505 

160000 

2623 

668 

i93 

5 

179-93 

5-6i 

n-73 

65.8 

0.0622 

156300 

2375 

669 

253 

5 

180.00 

5-64 

11.76 

66.3 

0.0608 

152300 

2297 

638 

ISI« 

6 
8 

180.00 
180.00 

4-50 
4-5° 

n.  60 
"•59 

?::(»«•- 

(  0.0670  1 
(  0.0750  i 

200000 

I9l6 

639 

IS}". 

7 
5 

180.00 
180.00 

4-52 
4.49 

11.66 
11.62 

?:;K» 

i  °'°A45  i 

/  0.0670  ) 

212000 

2021 

640 

131- 

7 
5 

180.00 
180.00 

4-53 
4-52 

"•59 
"•59 

£?!••>- 

i  0.1060  | 
1  0.0955  } 

149000 

1419 

642 

ISSl* 

6 
7 

179.98 
179.98 

5-57 
5-58 

ii.  61 
n.  61 

&K« 

(  0.0770) 
(  0.0390  j 

215000 

1661 

643 

IS!*- 

{'S 

179.92 
179.92 

5-65 
5-6i 

ii.  61 
11.62 

&SI-3- 

(  0.0440  | 
(  0.0600  ) 

261000 

J995 

644 

i3!«* 

i; 

179.96 
179.96 

5.60 
5-60 

11.63 
11.62 

&I-3-- 

|  0.0596  | 
I  0.0690  ) 

257800 

1980 

648 

l=Si«* 

\- 

180.00 
180.00 

5.60 
5-6i 

11.72 
11.72 

SSI-*.. 

|  0.0590  | 
(  0.0700  J 

268000 

2042 

649 

l:Ui»' 

9 
4 

180.00 
180.00 

5.60 
5-61 

11.71 
11.74 

§:$!•*•> 

(  0.0600  | 
J  0.0705  \ 

277000 

2107 

650 

lisi* 

12 

9 

180.00 
180.00 

5-6i 
5-61 

"•75 
11.71 

§:?!"••' 

\  0-0530  | 
}  0.0885  i 

24OOOO 

1824 

645 

laf* 

5 
7 

179.97 
179.97 

5-59 
S-6o 

"•59 
ii.  60 

§:;!•-« 

|  0.0560  | 
/  0.0540  i 

263200 

2028 

646 

!£!«» 

6 

12 

180.00 
180.00 

5-59 
5.60 

IT.  61 

11.62 

&?i'3°"> 

0.0493  | 
0.0620  5 

249000 

i9J5 

647 

151* 

" 

1*3 

180.00 
180.00 

5.62 
5-62 

11.62 
11.62 

§:!}•*"• 

0.0630 

/  0.0700  ) 

248000 

1899 

678 

IS!*6 

16 
8 

179.94 
179.94 

5.58 
5-57 

11.47 
"•45 

tll""* 

0.0529) 
0.0642  ) 

245500 

1921 

679 

I3I» 

11 

180.00 
180.00 

5.62 
5.62 

11.76 
11.72 

fi.l\  — 

0.0664  | 
0.0705  \ 

249000 

1886 

680 

IS!* 

ii 

180.00 
180.00 

5.60 
S-6i 

11.72 
"•73 

g:S!-3-« 

0.0650  1 

0.0495  $ 

278000 

2116 

663 

|3|«« 

it 

180.00 
180.00 

5.60 
S-63 

"•75 
"•75 

S:S!-3- 

0.0621  ) 
0.0657  i 

300000 

2273 

676 

Isil* 

(  12 
<      ^ 

179.94 
179.94 

5.60 
5-6i 

11.71 
"•73 

g:t|'3..4 

\  0.0530  1 
1  0.0593  i 

274500 

2089 

677 

IS!** 

i      ^ 

i    6 

180.00 
180.00 

5-6i 
5-68 

11.72 
11.72 

§:Jl«» 

j  0.0551  1 
/  0.0625  j 

255000 

<945 

COMPRESSION  OF   WHITE  PINE. 


663 


COMPRESSION    OF    WHITE    PINE.-  Concluded. 
SINGLE  STICKS  AND  BUILT  POSTS. 


No. 
of 
Test. 

Weight. 

_c 

22  So 
w^  c 

Dimensions  of  Post. 

Sectional 
Area. 

"ft 

c3 

Ultimate  Strength. 

M 

| 

Hi 

Q 

Actual. 

£• 

Ibs. 

in. 

in. 

in. 

sq.  in. 

in. 

Ibs. 

(175 

(18 

180.00 

4-52 

11.62 

52-5) 

I  0.0460  ) 

690 

<  226    600 

» 

180.00 

5-56 

ii  .70 

65.0}  169.3 

I  o.o58o[ 

310000 

1831 

'  T99 

(18 

180.00 

4.46 

11.62 

51-8) 

(  0.0480  ) 

% 

(164 

(    9 

179.98 

4-48 

ii  .60 

52.0) 

(  0.0526  ) 

691 

<  197    520 

>  I4 

179.98 

5-56 

1  1.  60 

64.5  J  168.2 

]  0.0430  } 

372500 

2215 

(i59 

(  12 

179.98 

4-45 

ii  .61 

51-7) 

(  0.0390  ) 

(248 

(13 

i77-25 

5-62 

1  1.  60 

65-2) 

(  0.0580  ) 

692 

{  153     6l4 

12 

4-50 

ii.  60 

52.  2  >  l82.2 

I  0.0641  \ 

363000 

1992 

(213 

(    5 

177-25 

5-6o 

ii-57 

64.8) 

(  0.0768  ) 

(  I5I 

(ii 

180.00 

4-5° 

ii.  60 

S2.2) 

(  0.0460  ) 

687 

J2i8    536 

8 

180.00 

5-58 

ii  .62 

64.8       169.4 

o.o587[ 

325500 

1919 

(  167 

(    9 

180.00 

4-52 

11-59 

52.4) 

(0.0533) 

(176 

(10 

180.00 

4-5° 

ii  .60 

52-2) 

(  0.0565  ) 

688 

}  236     575 

10 

180.00 

4.62 

1  1.  60 

65.2  >   169.6 

j  0.0645  J 

306000 

1804 

(163) 

(ii 

180.00 

4-50 

1  1.  60 

52.2) 

(  0.0703  ) 

(188 

(    7 

180.05 

4-48 

n.  60 

52.0) 

{  0.0510  ) 

689 

{  203     550 

11 

180.05 

5-6o 

11.62 

65.1  >  168.7 

o.o66oj 

340000 

2015 

(i59 

I    9 

180.05 

4.46 

"•57 

51-6) 

(  0.0789  ) 

fi93l 

r  s 

179-95 

4-47 

ii.  60 

5I-91 

{0.0700"! 

681 

38  * 

U57J 

1 

I   9 

179-95 
179-95 
179-95 

4-52 
4.48 
4-51 

11.65 
11.65 
11.65 

52-sJ 

0.0714  1 
0-0531  [ 
0.0762  J 

362000 

1734 

fi6i-| 

f    7 

180.00 

4-5° 

11.63 

52-31 

{0.0612") 

682 

!:§r 

14 
\l 

180.00 
180.00 
180.00 

4-5° 
4-49 
4.50 

ii  .64 
ii.  60 
11.63 

S'sJ 

0.0546  1 
0.0542  f 
0.0916  J 

414000 

1980 

f  169") 

fio 

180.03 

4.46 

11.63 

{0.0570^ 

683 

\l\l\rt 

J" 

1    9 

180.03 
180.03 

5.62 
5-6o 

11.69 
ii  .70 

234-9 

0.0530  I 
0.0494  f 

501000 

2133 

Ii8ij 

I   7 

180.03 

446 

II.  01 

o.osgoj 

ri46-| 

fio 

180.00 

4-5° 

11.64 

52-41 

f  0.0664! 

684 

180.00 
180.00 

5-64 
5-6o 

ii-59 
11.58 

J  0.0610  ! 
I  0.0548  f 

529000 

2255 

li66J 

1  10 

180.00 

4.48 

n.  61 

52-  oj 

(,o.05i4j 

f  145^ 

fto 

180.00 

4-5° 

1  1.  60 

62'2] 

(0.0645! 

6*5 

ixo 

1  12 

180,00 
180.00 

5.63 
5-6i 

11.62 
11.62 

65.2  J  234' 

0.0650  1 
0.0506  f 

430000 

1831 

Ii66j 

I  9 

180.00 

4.48 

11.62 

52.0] 

0.0546J 

686 

fi45l 

II* 

UsoJ 

(S 
I'l 

180.00 
180.00 
180.00 
180.00 

4.48 
4-50 
4-5° 
4-52 

ii.  61 
11.56 
11.36 
ii.  61 

52.  oj 
52-5J 

To.  0500"! 
J  0.0315  ! 
|  0.0543  f 

395000 

1903 

664 


APPLIED  MECHANICS. 


u 

S 

T3  0      • 

C    3      J-    'O 


.s  «  1  g  g 


iifilil 

.^^^ 


Q       fn  Q 


•-<  "-"  Ost^oo  O\T!-VQ  OVO  rot^ooo 
1OVO  ON^Oi-i  rOi-iVO  '-OO\fOLOO  "^ 
vOi-ir^cxDvOr^w'-iONNfONO^O 


MOO     O\i-c      >-> 


^ 


<& 


a. 


£  S 


\OvOvO\OvovOvO 
unio^^u^vo>-ou^ 


COMPRESSION  OF   YELLOW-PINE  POSTS. 


665 


*  . 

13  >^ 

Mil 

•r  fe    M  'C 


.s  s  -  «   «  .s  i  .s  S  .s 

oooo    g    t^"->^5    o 


T3 


5      3      3      3      3 


t^ro 
ONf^ 


N         N         HH         O 

n    vo     N    vo 
^"    ro    fO    CO 


oooo 

N      fO 


e-  oq    in  oq    ix  ON  q  oq    ~    N    woqoq    q\roq\vqvq    mvq    q 
•"•    o\   *+    GNI-<     ^O    ONOOOOOO    »o  »o  **1  *O  10  *o  ••^j-   to   LO  10 


*?  T  ^    T  T 


T}-  \O     •*  vo 


w      N      wOOVO 


0000     r^OvO     O     fOTfroO     OOO     O     <OTJ 

C'N      N      <T)fOCOr?rP'1"T*''f'~'      X      "      "      "i 

—   dvdvdvd    d    d    dvd^ovdodoood    d    d 


:   -  8  - 

d    d    •*  -4-  "$•  06 


NNNNNNMNN 


I»J 


oo 
<«»ON 

£00     H« 


ooqqq^o^qqqp 


666 


APPLIED   MECHANICS. 


o  ^ 

4->     ^3 

O    ^ 
.51 


S    "« 
e     C 

II 


11 


3    ,N 
"     O 


.15 

&  Q 


-^^3^13 
fe     Q  fe     Q 


s 


•5         « 

;- 


ro  O\  vO  H-I  »-O 
t^>.  »—  '  ON  O  OO 
O  "^~  t^  t^>.  CN 


I-HOO     t^- 

i-iOi-i 


N     TJ-    r>.    "~>   O    LOOOO     o     Lo^f 
OOOOQtN.NNNO\i-iOOO 

rorONNNNNONNM 


.2  S  "*    "-O  ^O     ^o    ^    to   ^o    ^*    ^-O    O     O    O     r^    t^    r^    ON    f'l     ^O    ro    T^-    co 

.-^   r^  NO    r^   r^   r^>.   t^    t^.   t^»   t^*   r^  vo   CO   oo   00     O     O     O     O     ^o   ro 

fOONOoooo    t^rovO    t^»o^-Q    r~.O    "i   w     TJ-M     «-oo 
t^    "^-  oo     'o  O     i^.    r^.    t^s.    ^-    ^-    ON  NO     ^o   ^o  oo     ly")  vo    NO     r^i     ci 

PL,  72  .S*  ..............       .       . 

TJ-  NO     ^o  00     ^    O     O 
c/3  _e  c'WNMnc<rjN 

—  ooodddd^^Tt-oo'odoo'dddddd^dd 
£          $;  NO  vo    o    o    o 

r|  s  a '  '   '  2  s 

<j      o l?i 

qLoqqqqqqqqqqqqqqqqqq 

'S  £   OO     ^O  NO     *-<     ON    O     O    NO 

^ 

OOONO^      N      NfOrJ-t^OOON^      N      CO^OOON<3w'-i 
^^^^^^^^^^^^^^^-OO^v^vS- 


COMPRESSION  OF    YELLOW-PINE   POSTS. 


667 


.1 


<!,       O 


13     " 


"         •-' 


in    iJ     ^    w     en 

-  §  .s  3  3 
s  8 1  5  ^  s 


« 

VO 

O\ 


c-   co 


o      q 
%     a 


668 


APPLIED  MECHANICS. 


COMPRESSION    OF    YELLOW    PINE. 
SINGLE  STICKS  AND  BUILT  POSTS. 


No. 
of 
Test. 

Weight, 
in  IBs. 

11 

<;    f2 

Dimensions  of  Post. 

Sectional 
Area. 

ft 

Ultimate  Strength. 

t 
J 

i 

! 

1 

J* 

in. 

in. 

in. 

sq.  in. 

260     ) 

(  180.05 

4.09 

"-35 

46.04    ) 

1142200 

3065 

673a 

J3 

ii 

<      20.00 

4.01 

4.01 

16.08    J 

0.0370 

94000 

5846 

673^ 

7$   ) 

(     20.00 

3-95 

3-97 

15-68   ) 

99600 

6352 

674 

233 

17 

iSo.OO 

4-5i 

n.  60 

52-30 

0.0492 

131500 

2515 

675 

194 

22 

iSo.OO 

4-34 

n.  60 

50-30 

0.0490 

I2I200 

2410 

490 

269 

- 

iSo.OO 

5-05 

12.  IO 

6i  .  10 

- 

230000 

3764 

670 

309 

12 

iSo.OO 

5-65 

11.74 

66.30 

0.0455 

205900 

3106 

489 

287 

- 

180.08 

5-85 

12.05 

70.50 

- 

250000 

3546 

654 

ISI« 

1" 

1    6 

iSo.OO 
iSo.OO 

5-63 
5-62 

"'.71 

65-9 
65-9 

I3I-8 

(  0.0418  | 
1  0.0540  i 

470000 

3566 

655 

isl*" 

\l\ 

179-93 
179-93 

5-64 
5-63 

11.72 
JI.72 

66"  }'3»-» 

(  0.0292  ) 
I  0.0315  i 

580000 

4387 

6S6 

i  Si  5*9 

!» 

iSo.OO 
iSo.OO 

5.61 
5-61 

11.71 
11.71 

JflJ 

I3I-4 

(  0.0368  ) 
/  0.0466  j 

480000 

3653 

651 

i  275!  *>7 

11 

iSo.OO 
iSo.OO 

5-58 
5-58 

11.71 
11.71 

otli130'6 

50.0620  ) 
0.0514  I 

360000 

2756 

652 

j|£|«o 

(16 
(16 

iSo.OO 
iSo.OO 

5.58 
5.58 

11.70 

11.71 

65-3 
65-3 

130.6 

i  0.0395  { 
(  0.0360  $ 

588500 

4506 

653 

J3"!621 

(16 

iSo.OO 
iSo.OO 

5-63 
5-59 

11.71 
11.68 

65-9 
65-3 

I3I-2 

j  0-0559  \ 
1  0.0550) 

436600 

3328 

657 

l^!659 

!  13 

179.96 
179.96 

5.63 
5-64 

11.72 
11.71 

66.0 
66.0 

132.0 

{  0.0375  \ 
1  0.0305  ) 

SSOOOO 

4394 

658 

l^i651 

11 

179.98 
179.98 

5-59 
5-59 

11.71 

11.72 

65.5 
65-5 

I3I.O 

(  0.0320  ) 
1  0.0436  j 

448000 

3420 

659 

i'Si6^ 

n 

iSo.OO 
l8o.OO 

5-6i 

"-73 
"-73 

65.8 
65.8 

I3L6 

(  0.0312  ) 

I  0.0372  } 

6OOOOO 

4559 

660 

i  Si  577 

i°i 

180.03 
180.03 

5.61 

5-63 

11.22 
I1.24 

62.9 
63-3 

126.2 

j  0.0325  I 
I  0.0400  > 

510000 

4041 

661 

5  3™  J584 

11 

iSo.OO 
iSo.OO 

5.66 
5-60 

11.70 
11.72 

66.2 
65.6 

I3L8 

50.0410  ) 
0.0365  i 

410000 

3i" 

662 

\l&\$7* 

{•I 

l8o.OO 
iSo.OO 

5.61 
5.61 

"-75 
"•75 

65-9 
65-9 

I3I.8 

(  0.0540  | 
<  0.0500  ) 

388000 

2944 

(242) 

(ii 

179.97 

4-50 

ii.  61 

52.2 

10.0320  ) 

693 

j  276    774 
(256) 

15 
(23 

179.97 
179.97 

5-5o 
4-49 

11.56 
11.62 

63-6 
52.2 

168.0 

0.0325  | 
0.0148) 

564000 

3357 

(i93) 

(i? 

l8o.OO 

4.50 

"•35 

51-! 

(  0.0500  j 

694 

isi691 

24 
(10 

iSo.OO 
iSo.OO 

5-59 
4.46 

11.36 
"•35 

63-5 
5O.6 

165.2 

(0.0610) 

5000OO 

3027 

(207) 

18 

iSo.OO 

4-49 

"•35 

51-0 

(  0.0429  1 

695 

<  290  >  743 
(246) 

16 

iSo.OO 
ISO.OO 

5-20 
4-50 

"•34 
"•35 

5I.I 

161.1 

<  0.0290  > 
(0.0410) 

474000 

2942 

COMPRESSION  OF   YELLOW  PINE. 


669 


COMPRESSION    OF    YELLOW    PINE.—  Concluded. 
SINGLE  STICKS  AND  BUILT  POSTS. 


Weight, 
in  Ibs. 

"8       . 

•S*s  « 

No. 

j.a 

Dimensions  of  Post. 

B  Jj 

Ultimate  Strength. 

of 
Test. 

!i* 

^ 

Sectional 
Area. 

gjs'gjg 

ils' 

2  2  & 

ti 

*5 

•s 

ft  o  m    • 

1 

. 

£O.S 

a 

r2 

Pi 

u 

H*  II    C/2 

| 

(/)   O* 

<    « 

3 

^ 

Q 

3 

< 

.flW 

in. 

in. 

in. 

sq.  in. 

(217 

(25 

180.00 

4-51 

11.24 

50-7) 

!  0-0337  ) 

696 

253 

660 

11 

180.00 

5-50 

11.23 

61.8(162.8 

0.0567  j 

480000 

2948 

(190 

(15 

180.00 

4.48 

11.23 

50-3) 

0.0715) 

(242 

(is 

180.00 

4-52 

ii  .60 

52-4) 

f  0.0240  1 

697 

249 
(215 

706 

12 
(l5 

180.00 
180.00 

5-43 
4-47 

n.  60 
11.58 

63.0  >  167.2 
51-8) 

j  0.0330  J 

(  0.0336  ; 

540000 

3230 

(224 

(15 

179.94 

4.46  j     n.  60 

51-7) 

(0.0290) 

698 

255 

775 

8 

179.94 

5-53 

11.70 

64.7*168.6 

j  0.0385  [ 

544000 

3227 

(296 

(IS 

179.94 

4-5° 

"•59 

52.2) 

(  0.0341  ) 

488 

911 

~ 

180.20 

J6.88 
|6.7a 

15-75 
15-75 

SUI-*- 

- 

700000 

3268 

On  page  668  will  be  found  a  series  of  tests  of  spruce  columns 
made  in  the  Laboratory  of  Applied  Mechanics  of  the  Massachu- 
setts Institute  of  Technology,  these  columns  having  their  ends 
resting  against  the  platforms  of  the  testing-machine.  On  the 
same  page  will  be  found  also  a  series  of  tests  made  at  the  same 
place  on  timber  columns  with  one  end  resting  against  a  timber 
bolster. 

A  perusal  of  this  table  will  show  a  great  decrease  in  strength 
due  to  the  presence  of  the  bolster. 

The  four  diagrams  on  the  left  of  page  669  represent  all 
the  tests,  with  central  load  and  no  bolsters,  of  the  four  woods 
named,  the  abscissae  being  ratios  of  length  to  least  diameter 
and  the  ordinates  breaking-strengths  per  square  inch. 

By  whatever  curve  we  may  attempt  to  represent  the  values 
to  be  used  in  practice,  it  should  pass  nearly  through  the  lowest 
results,  as  the  timber  was  all  of  at  least  fair  quality.  It  is  also 
evident  that  up  to  a  certain  ratio  of  length  to  diameter,  not 
far  from  25,  the  strength  is  not  affected  by  the  length,  and 


670 


APPLIED    MECHANICS. 


COMPRESSION  OF  SPRUCE  COLUMNS. 

Remarks. 

Deflected  diagonally. 

"  Crushed  5  feet  from  platform, 
downward.  "  at  centre, 
diagonally.  "  4i  feet  from  platform, 
horizontally.  "  i'  3"  from  respective  ends, 
downward.  at  knot  18"  from  end. 
Crushed  i'  from  platform. 
6" 
3' 

*;? 

Deflected  horizontally.  Crushed  at  end.  Part  of  column 
tested  December  8,  1894. 
Deflected  diagonally.  Crushed  at  centre. 
Crushed  near  one  end. 

COLUMNS  TESTED  WITH  A  BOLSTER. 

Manner  of  Failure. 

1           II 

"o                    "0*0 

•O                             &                  & 

o               22 

•0.0                        -O^i             T3 

go-  s   5   =    £o;   3    g 

'     C                        'i-i  G             "J.. 

2  £*  '  *  *  w'g3  *  2 
•atSs  3  2  =  etS:  r  -g 

S  ^                        ^^             rt 

r^s-  :  =  s  "is*  :  ^ 

"a"E              ^"5.        a 

C/3C/)                      Uc/2            C/3 

1|| 

U                             S 
"3222^"«2222^ 

i  >*-i  c 

<U  O  £ 

llg 

u                  <u 
^3                ^  3 

5^       O."                      I?    CL-~ 
C/5                 Oc/5 

-nsBjg    jo    snjn 

-P°W    p3UBD    -M 
-uocatnoD  'UIT245S 

Ol  SS3J1S  JO  OU'E'JJ 

Jooooooooooooo      o    • 
OOOOO^OOOO    t^oo    O         O     • 
\o  moo  p)N"->ot^m-<rpimo        o    • 

o  Ko?  «  *2  Inoo"  5-'  N  oo*  O 

H^HIillllHi'  •§  : 

•uj  'bg 
vj*         j3d  -sqi 

tO    K*« 

IsllEfllltpH  II 

MNOOOOOOOOO 
pi   0   p-  0   -   m  tCvo^  O^oo^  ^_ 

a  a 

'Z  w 

88888888888888     88 

O    O    O    fO  O    O    ""i  i^OO    t^  O  O    **t&>         O    O 

8QOQQQQQOOC 
OOOOOOQOOC 

\?,  §.  Z£  &  ooo  o  R  R  o^  KcS       oo 

S"&S  ^o  9,  ?.  ?  5"  ^1 

•apiS  isuaq  01 

O  f^  O  O  moo  f^.  m  o  -o-oo  o  ^o  ^o       •<*••*• 

O  00    t^  O    -    t^  rr.  o  r^  in  c 

.,MV,-on»S 

e  8  S,  8  8  S  8  8,  §  S  S  "8  ?  o  vS     88 

O    OOO    000    OOO   -*P)    O^C 

crvS-S  88^^o^8S^vg^^S     88 

^cSSoo^^ooSS 

i 

-.S'ooS^^S^^^^Sof?^    88 

"m  c^oo  8  oo  tC  tCvo1  &  8  ^' 

.£0000  6  600  N  csoooooo  t^  ooo  o       O  O 

O   O   ON   OOt^OOO    C 

3 

g        -VPIAV 

8100  O  mooo  mmm  irioo  10  m       00 
NOOi-OOOMMW    ^00    N    -•           OO 

.S  oo  co  0  o  oo  oo  tvoo  oo  oo  t^.  r^-oo  d        O  O 

Jlo^o^oood 

a 

c'  0  0  0  0  8  £00  w1  8  <mtm  1C  o_  c?       88 
worxovo^cooowo       gg 

^    t^OO  *OU->MOOOt>«Of.lAt~O          DP) 

O  w  iriN'O  o  t^O  mmt*-- 
*t~\O    O    N    uivc  vo    O    0s  i-1    u" 

•,,3u»T  »Sno 

.g^^^-.-.      O^^^^^      g         g 

a^&a&^agssg 

.c  2  "                                    22 

•»*.* 

.8'8*&5\g8<g'88RS58  J8      1  : 

5-  moo1  pToo  oo  looo'o?  tCvc 

NNmm      KWIHM                »N 

.W0 

o  o  i-  m  to  -*vo  vo  ,    .   ^   ^  ^   oo       m 

o  «                 .                                      «  o 
§5  -   ••  «     fa  •••»».»-     oooc'XJ 

ltJ3   S     ^    :   3    S    8*3   3 
0         55                 Q 

COMPRESSION  OF    YELLOW  PINE. 


67I 


672 


APPLIED    MECHANICS. 


hence  this  part  of  the  curve  should  be  a  horizontal  line,  its 
ordinate  being  about  3500  for  yellow  pine,  2COO  for  white  pine 
or  spruce,  and  3000  for  white  oak. 

For  larger  ratios  than  25  the  ordinate  decreases,  and  might 
well  be  represented  by  some  curve  similar  in  character  to  those 
on  page  42$. 

The  right-hand  diagrams  are  due  to  Mr.  E.  F.  Ely,  and 
represent  the  lowest  results  of  the  Government  tests  on  yellow 
pine  and  white  pine,  and  also  the  values  he  proposes  for  use  in 
practice.  Mr.  Ely's  diagrams  are  represented  by  the  following 
rules:  Let  A  =  area  of  section  in  square  inches,  f  =.  constant 

given  in  the  tables  following,  —  =  ratio  of  length  to  least  side 

of  rectangle. 

Then  breaking  strength  ==  fA. 


White  1 

inc. 

Yellow 

Pine. 

/ 

/      1 

/   - 

/ 

r 

r 

o  to  10 

25OO 

o  t  15 

4000 

10  to  35 

2OOO 

15^  30 

3500 

35  to  45 

I5OO 

30  to  40 

3000 

45  to  60 

IOOO 

40  i  45 

2500 

45  '  50 

2000 

1 

50  l  60 

15OO 

In  the  case  of  spruce  and  white  oak,  if  it  is  desired  to  ap- 
ply the  results  to  greater  ratios  of  length  to  diameter  than  those 
tested,  a  similar  reduction  can  be  made  in  the  value  of  /to  that 
which  takes  place  here  in  the  case  of  white  and  yellow  pine. 

§  238.  Factor  of  Safety. — Whereas,  in  the  case  of  iron 
bridge-work,  it  is  very  common  to  use  a  factor  of  safety  4,  the 
apparent  factors  of  safety  that  have  been  used  and  recom- 
mended for  timber  have  varied  very  greatly,  and  naturally  so,1 
because  the  values  assumed  for  breaking-strength  have  been  so 


TRANSVERSE   STRENGTH  OF   TIMBER.  673 

very  variable,  and  while  some  have  advised  the  use  of  apparent 
factors  of  safety  greater  than  4,  nevertheless  most  of  the  build- 
ing laws  only  require  an  apparent  factor  of  safety  3,  while 
making  use  of  values  of  breaking-strength  deduced  from  tests 
of  small  pieces. 

In  view  of  the  above  facts,  it  is  true  that  the  values  of  work- 
ing-strength used  in  many  cases  have  been  very  near  the  actual 
breaking-strength ;  and,  indeed,  it  would  be  impossible  to 
recommend  any  suitable  factor  of  safety  to  be  used  with  re- 
sults derived  from  tests  of  small  pieces.  But  if  the  true  values 
of  the  breaking-strength  as  derived  from  tests  of  full-size  pieces 
be  used,  it  would  seem  to  the  writer  that  a  factor  of  safety  4 
will  be  sufficient  for  most  ordinary  timber  constructions ;  i.e., 
that  we  should  use  for  working-strength  per  square  inch  one- 
fourth  of  the  breaking-strength  per  square  inch.  In  the  case 
of  mill-work,  and  in  other  cases  where  there  is  the  jarring  of 
moving  machinery,  it  is  advisable  to  use  a  somewhat  larger 
factor.  This  same  reasoning  will  also  apply  to  the  case  of 
beams  bearing  a  transverse  load,  where  they  are  designed  with 
reference  to  their  breaking-weight. 

§  239.  Transverse  Strength  of  Timber.— The  table 
of  Rankine,  already  given,  represents  the  values  of  modulus 
of  rupture  as  obtained  from  small  specimens.  Other 
values,  not  differing  essentially  from  these,  are  given  by  Hat- 
field,  Laslett,  Thurston,  Trautwine,  and  others,  all  based  upon 
tests  of  small  pieces.  Confining  ourselves  to  tests  of  full-size 
pieces,  we  find  an  account  of  a  set  of  tests  attributed  by  D.  K. 
Clark,  in  his  "  Rules  and  Tables,"  to  Edwin  Clark  and  C.  Gra- 
ham Smith,  The  results  are  given  below,  and  it  will  be  seen 
that  they  are  very  much  below  those  given  by  experimenters 
on  small  pieces. 


674 


APPLIED  MECHANICS. 


Kind  of  Timber. 

Breadth 
and 
Depth. 

Span. 

How 
Loaded. 

Breaking- 
Weight. 

Modulus 
of 
Rupture. 

in. 

ft. 

American  red  pine 

I2.O   X    I2.O 

15.00 

Centre 

33497 

5238 

"           "       " 

I2.O   X    I2.O 

15.00 

" 

29908 

4680 

<«                       <<             H 

6.0   X      6.0 

7-5° 

" 

7370 

4608 

Memel  fir      ... 

13-5   X    13-5 

10.50 

Distributed 

68560 

5274 

"        " 

13-5   X    13-5 

10.50 

" 

68560 

5274 

Baltic  fir  .... 

6.O   X    I2.O 

12.25 

Centre 

J9r45 

4878 

«       « 

6.O   X    I2.O 

12.25 

" 

23625 

6O2O 

Pitch  pine     .     .     . 

6.O   X    I2.O 

12.25 

" 

23030 

5868 

«         « 

6.O   X    I2.O 

12.25 

« 

23700 

6048 

"        "        ... 

14.0  x  15.0 

10.50 

" 

134400 

8064 

"        "... 

14.0  x  15.0 

10.50 

" 

132610 

7956 

Red  pine  .... 

6.0   X    12.0 

12.25 

" 

16800 

4284 

«        « 

6.O   X    I2.O 

12.25 

" 

19040 

4860 

Quebec  yellow  pine 

14.0  x  15.0 

10.50 

Distributed 

68600 

4I22 

"           "         " 

14.0  x  15.0 

10.50 

" 

68600 

4T22 

«           <«         « 

14.0  X    15.0 

10.50 

Centre 

85792 

5[48 

"      "     " 

14.0  x  15.0      10.50 

" 

76160 

4572 

Two  tests  by  R.  Baker  are  also  mentioned  by  D.  K.  Clark. 

Bauschinger  also  made  quite  an  extensive  series  of  tests  of 
German  woods,  an  account  of  which  will  be  given  later  on. 

A  great  many  tests  of  the  strength  and  stiffness  of  full- 
size  beams  of  spruce,  yellow  pine,  oak,  and  white  pine,  both 
under  centre  loads  and  distributed  loads,  have  been  carried 
on  in  the  Laboratory  of  Applied  Mechanics  of  the  Massa- 
chusetts Institute  of  Technology.  Tests  have  also  been  made 
upon  the  effect  of  time  on  the  stiffness  of  such  beams,  also 
on  the  strength  of  built-up  beams,  and  of  floors  and  fram- 
ing-joints, all  full  size.  A  summary  of  the  results  obtained 
will  be  given,  and  conclusions  drawn  as  to  the  proper  values 
of  the  modulus  of  rupture  and  modulus  of  elasticity,  etc.,  to 
be  used  in  practice. 


TRANSVERSE  STRENGTH  OF   TIMBER. 


675 


SUMMARY   OF    THE   TESTS. 


The  tests  recorded  may  be  divided  into  six  classes :  — 


i°.  Spruce  beams. 

2°.  Yellow-pine  beams. 

3°.  Time  tests. 


4°.  Oak  beams. 

5°.  White-pine  beams. 

6°.  Framing-joints. 


i°.  Spruce  Beams.  — Before  giving  a  summary  of  the  tests 
made  in  this  laboratory,  I  will  insert  some  of  the,  moduli  of 
rupture  and  moduli  of  elasticity  given  by  different  authorities. 

Moduli  of  rupture  are  given  as  follows  :  — 


Maximum. 

Minimum. 

Mean, 

Hatfield     .... 

12996 

7506 

9900 

Rankine      .... 

12300 

9900 

IIIOO 

Laslett  

9707 

7506 

9045 

Trautwine  .... 

- 

- 

8lOO 

Rodman     .... 

- 

- 

6l68 

Hatfield's,  Laslett's,  Trautwine's,  and  Rodman's  figures  are 
from  their  own  experiments.  Trautwine  advises,  for  practical 
use,  to  deduct  one-third  on  account  of  knots  and  defects,  hence 
to  use  5400.  The  tables  show  the  values  obtained  in  these 
tests,  and  I  will  add  a  recommendation  as  to  the  values  of 
modulus  of  rupture  and  modulus  of  elasticity  suitable  to  use  in 
practice. 

As  a  result  of  the  tests  thus  far  made  in  my  laboratory, 
it  seems  to  me  safe  to  say,  if  our  Boston  lumber-yards  are 
to  be  taken  as  a  fair  sample  of  the  lumber-yards  in  the  case 
of  spruce,  —  if  such  lumber  is  ordered  from  a  dealer  of  goo*1 
repute,  no  selection  being  made  except  to  discard  that  which  is 
rotten  or  has  holes  in  it,  —  that  3000  Ibs.  per  square  inch  is  all 
that  could  with  any  safety  be  used  for  a  modulus  of  rupture, 
and  even  this  might  err  in  some  cases  in  being  too  large ; 
(2°)  that,  if  the  lumber  is  carefully  selected  at  any  oce  lumber- 


676  APPLIED    MECHANICS. 

yard,  so  as  to  take  only  the  best  of  their  stock,  it  would  not  be 
safe  to  use  for  modulus  of  rupture  a  number  greater  than  4000 ; 
and  if  we  required  a  lot  of  spruce  which  should  have  a  modulus 
of  rupture  of  5000,  it  would  be  necessary  to  select  a  very  few 
pieces  from  each  lumber-yard  in  the  city.  With  a  factor  of 
safety  four,  we  should  have  for  greatest  allowable  outside  fibre 
stress  in  the  three  cases  respectively,  750,  1000,  and  1250. 

The  modulus  of  elasticity  (i.e.,  that  determined,  from  the 
immediate  deflections)  was:  maximum,  1588548;  minimum, 
897961;  mean,  1332500. 

As  will  be  explained  when  the  results  of  the  time  tests  are 
given,  if  by  means  of  the  ordinary  deflection  formulae  we  wish 
to  compute  the  deflection  which  a  spruce  beam  will  acquire 
under  a  given  load  after  it  has  been  applied  for  a  long  time, 
we  should  use  for  modulus  of  elasticity  in  the  formulae  not 
more  than  one-half  of  the  values  given  above,  or  about  666300. 


TRANSVERSE   STRENGTH  OF   TIMBER. 


677 


SPRUCE   BEAMS. 


c 

c.5  c 

u 

•a 

c_  a3 

IT.     'S. 

u 

•o 

3  a 

j5  & 

4^     U  *"* 

£ 

rt 
o 

*£ 

w-1  . 

J~  S  o* 

8 

•o 

§5 

o  2 

u 

* 

CJ 

*  c  e 

|-~~. 

l^g. 

Manner  of  Breaking. 

Q 

.c  a 

s£ 

=  bi 

3 

^   z  ^~* 

"3.-  u 

'H  >,  & 

3  3 

c  2r 

rt 

•c  -  a* 

T3   ;j  4) 

Q 

•  —  Q 

.SEW 

§.s 

V 

O   ~  C/3 

"rt  "5J  •"•  « 

55 

£ 

O 

JS 

cc 

2 

I"' 

u 

inches. 

ft.    in. 

Centre 

Ibs. 

26 

1,2^7  7OO 

181 

Crushing  and  tension. 

4 

2x9 

6    7* 

7,322 

5,389 

1,067,900 

301 

5 

2X12 

" 

5,586 

5,237 

Tension. 

c^i 

2X12 

7     ° 

44 

8,982 

-*" 

6 

2fx    9 

6     8 

4« 

7,586 

4.082 

938,500 

230 

" 

7 

3    x    9 

4     o 

44 

1  1,  086 

3,285 

308 

" 

8 

3    x    9 

10    o 

*k 

6,086 

4,508 

170 

*4 

9 

10 

3x9 
Si  x  » 

15     o 

20      0 

1! 

8,086 
6,586 

4-253 



141 
1  06 

Tension  and  crushing. 

Tension. 

208 

12 

I3JXI2 

16    o-j 

4*  ft- 
from  end 

}    7,585 

3,271 



19.6 

» 

Centre 

O     _      Q 

Crushi  ng. 

15 

ifx    6* 

7     o 

4^785 

7-562 



304 

Tenbion. 

16 

3    x    9 

6     8 

*' 

.    (jor 

277 

17 

6     8  \ 

4  points 

'  il'9" 

'    01 

46  q 

«i 

1  / 

3x9 

16"  apart 

r      ,  744 

4»y 

4L  J 

18 

30  x    i 

16    o-! 

4*  ft. 

(        _g 

218 

2O2 

it 

•  y      * 

I 

from  end 

) 

*5<i 

'9 

2X2 

14      o 

Centre 

4,404 

3.854 

1,482,600 

138 

44 

•20 

2X2 

14      o 

11 

5,  108 

4.469 

1,588,500 

1  60 

44 

2  L 

o  *  »  X      2 

14       o 

*  4 

8,627 

3,854 

I,l87,IOO 

137 

t4 

22 

3ix    2 

14       o 

44 

12,545 

5.666 

1,332,700 

Tension  and  shearing. 

23 

3*  x    ^ 

14       o 

44 

6,917 

2,995 

898,000 

"108" 

Tension 

24 
25 

2X9| 

14       o 
14      o 

!i 

8,927 
3,t98 

5.442 
4,139 

1,572,500 
1,460,600 

123 

Shearing. 
Crushing  and  tension. 

,6 

2j  X      2 

14      o 

" 

6,819 

4-339 

1,396,700 

Tension. 

27 

I  IB  X     O 

J4         0 

1C 

4,3o6 

1,355,9°° 

167 

" 

28 

4^  X      2 

18      o 

44 

8,829 

Jisce 

1,397,100 

^34 

4i 

2O 

3- 

4    x    2i 

3|-  x    2 

18      o 
18      o 

\ 

8,324 
7,721 

4,586 

5,559 

1,259,200 
1,231,500 

129 

Crushing  and  shearing. 

35 
36 

6x2 

2     X      if 

18      o 
7      2 

\ 

n,  188 
7,870 

4,196 
3,599 

1,347,900 

Shearing. 
Shearing  and  crushing. 

37 

4    x    if 

12          0 

4 

10,572 

4,i35 

169 

Tension. 

45 

16      4 

8,072 

111 

46 

3ia  x    2 

10         2 

Centre 

T3<772 

4,43" 

4,746 

1  jj 

Shearing. 

49 
60 

3*x    if 

4x2 

14      o 
17       4 

12  points 

12,076 
26,000 

5,878 
7.448 

1,461,000 

205 
406 

Tension  and  crushing. 

66 

4x2 

IS      8 

Centre 

14,576 

7,211 

1.336.200 

230 

Tension, 

70 

4i  x    2^ 

17       4 

12  points 

14,633 

3,748 

1.551,800 

20  5 

72 

4     x     2j\ 

17       4 

•• 

",333 

1.228,600 

177 

•' 

90 
ig6 

6x2 
3*8  x    ii| 

17       4 
19     10 

Centre 

26,100 
9.187 

5,  102 

6,037 

1.587,600 
1,513,400 

153 

Shearing. 
Crushing. 

'97 

355  x      I 

18      4 

44 

6,486 

4,757 

1,282.000 

120 

Tension. 

198 
199 

3lx    'J 
3*x    lit 

17      6 
19      6 

, 

10,178 
9,784 

5'993 
6,049 

1.529,000 
l,S  10.900 

170 

L 

200 

4    x    2f 

19     10 

4 

6,597 

3,845 

1  .  100,200 

IOI 

u 

201 

3*1  x    ,J 

18       4 

4 

10.958 

5,94t 

1,490,200 

191 

4* 

2O2 

3&X      231, 

18     10 

4 

10,077 

5-946 

1,475.100 

160 

*4 

203 

3  1  e  x     i  si 

19     10 

• 

11,184 

7,626 

1,850.600 

189 

Crushing. 

204 
205 

3ljx    i§j 

33z  x      2gj 

18       8 

18       8 

it 

10,970 
8,082 

7,066 
4,727 

1.470,400 
1,208,000 

188 

123 

Tension. 

297 

4X2 

15      5 

44 

6,025 

2,923 

1,010,000 

97 

•• 

298 

3*Jx    2 

9      8 

44 

J5,335 

4,700 

1,215,900 

299 

3*1  x     2 

9      8 

11 

4,484 

1,126,400 

232 

" 

6/8 


A  P r f  /  ED    ME CffA  NICS. 


SPRUCE    BEAMS. 


"3 

o' 
55 

Width  and 
Depth. 

Distance  be- 
tween Sup- 
ports. 

Manner  of 
Loading. 

PQ 

0.<n" 

|ls. 

rt  j2    • 

<«.S  cr 

^•^  o. 

"5  ai 

X-0     ' 

""•.a  «j 
J 

Manner  of  Breaking. 

inches. 

ft.    in. 

Ibs. 

300 

4   x  12 

9    6 

Centre 

19.040 

5,652 

1,173,000 

298 

Tension. 

301 

4   x  12^ 

9    6 

** 

19,041 

5,537 

1,210,000 

295 

14 

3°4 

4^x12 

" 

8,475 

3,852 

1,466,800 

Shearing. 

3°5 

4T8  X  12 

15    o 

*' 

9,779 

4,515 

1,345,600 

T52 

Tension. 

3°7 

418X12 

16    o 

44 

10,233 

4,594 

1,238,IOO 

156 

" 

308 

4!  X  12 

16    o 

*4 

12,186 

5,735 

1,466,700 

179 

309 

j  ? 

Sit  x  I2i 

16    o 

it 

9,283 

4.618 

1,067,300 

141 

Tension. 

4J?  x  12 

16    o 

it 

8,891 

4,184 

I,I56,IOO 

129 

" 

3r4 

3l  X  12 

16    o 

•4 

6,670 

3,442 

978,700 

102 

44 

316 

4    x  n| 

16    o 

ii 

11,885 

6,068 

1,290,300 

l84 

it 

3'7 

4r5Bxi2 

16    o 

N 

12,189 

5,638 

1,479,400 

175 

44 

4    x  nf 

16    o 

d 

11,875 

6,  1  1  3 

1,414,800 

191 

44 

3'9 

Si  X  12 

16    o 

it 

12,386 

6-393 

I.470.9QO 

201 

Crushing. 

320 

STB  x  IIli 

16    o 

4« 

5,386 

2,828 

1,092,600 

85 

321 

Sixni 

16    o 

(i 

13,086 

8,120 

I,8yQ,8oo 

Tension  and  shearing. 

322 

4     *  *2TB 

16    o 

d 

9,571 

4,639 

1,081,400 

145 

Tension. 

323 

Six  12 

15      0 

it 

9,170 

,585 

1,196,600 

154 

" 

325 

4     ^  *2g- 

16    o 

41 

8,175 

,003 

979,  3co 

128 

44 

AXI2 

15    o 

M 

4,665 

,187 

890,700 

74 

328 

Six  ii| 

i 

IO      O 

II 

12,375 

,522 

i  ,701,500 

203 

Shearing. 

33° 

3i  x  12 

15      0 

41 

6,075 

,037 

1,089,500 

103 

333 

4^  X  Ilyf 

J5    o 

41 

5.353 

,512 

849,500 

83 

336 

4     X  12^ 

44 

7,961 

3,655 

1,002,400 

124 

Tension. 

344 

3yf  X  I2g 

IS      0 

41 

10,761 

5,184 

1,369,100 

" 

348 

3!  x  i  if 

II 

13,162 

6,652 

1,689,600 

217 

" 

349 

3|x  n|| 

15      0 

II 

7,76o 

3.801 

1.116,300 

126 

Tension  at  knot. 

35 

3||X  ITj. 

15    o 

II 

10,464 

5.385 

1,382,000 

176 

Crushing. 

353 

3i  x  12 

IS      0 

44 

11,978 

5.989 

1,253,000 

200 

ii 

354 

4    x  121*8 

16    o 

II 

8,184 

4,049 

1,169,000 

127 

Tension. 

3lBx  IJIS 

15    o 

44 

7,760 

3,893 

1,233,400 

126 

" 

356 

3!  x  ii| 

15    o 

14 

Q  .961 

4,774 

1,464,700 

*57 

44 

359 

3    X9| 

17    o 

44 

4,74° 

4,958 

1,147,500 

1  20 

Compression  and  tension 

363 

4TB  x  I2& 

15      0 

4* 

10,170 

4,598 

1.285.100 

157 

14                       41               *i 

~7  X  12  * 

17    o 

44 

9,176 

4,980 

1,129,400 

366 

3    X9| 

17    o 

It 

5,540 

5.793 

1,411,200 

141 

Compression  and  tension 

367 

2$-  X  9| 

17    o 

*4 

1,630 

1,869 

1,211,000 

47 

Tension. 

3^|  x  12$ 

17    o 

14 

9,876 

5,217 

1,373,000 

157 

Tensionand  compression 

___ 

4    x  12 

17    o 

II 

10,789 

5»732 

1  7O 

372 

7    j 

12      0-j 

2  points  2' 
from  centre 

[    3,722 

854,100 

94 

Tension. 

373 

3   xgf 

12      0-j 

2  points  2' 
from  centre 

j-    9,595 

4,733 

1,126,500 

244 

Tensionand  compression 

374 

2|X9f 

12      O-J 

2  points  2' 
from  centre 

•10,767 

5,673 

1,593,000 

289 

Shearing,  j 

( 

£  at  centre 

377 

2lfX9l 

H 

i~3'  each 
from  centre 

•    4,626 

2,722 

1,072,200 

128 

Tension. 

378 

3|XI2* 

15    6-j 

2  points  2' 
from  centre 

17,259 

6,252 

1,720,000 

277 

Tension. 

380 

1  IS  x  9^| 

12      0 

Centre 

4,216 

5,086 

1,361,000 

176 

Tension. 

383 

3|  X  12^ 

16    o 

*' 

9,474 

4,840 

1,205,000 

I53 

2|X  10 

ii     6 

44 

6,739 

1,537,000 

244 

Compression. 

385 

2|  X  IO 

ii     6 

•• 

5,o°5 

3,604 

1,082,000 

Tension. 

387 
387" 

392 
I  395 

4    x  nf 

?i*Ks 

si*  x  nit 

16    o 
16    o 
ii     6 
16    o 

12      0 

ii 

9,869 

7.7°2 
15,813 
8,132 

4,735 
6,232 
5,945 
9,036 
3,339 

1,666,600 
1,442,800 
1,445,000 
2,132,400 

149 
191 

284 

Tensionand  compression 
Shear!  ng,  cr  ushin  g&  te  n  . 
Tensionand  compression 
Tension. 
Shearing. 

1  397 

4    xi2 

17    o 

" 

9,574 

5,084 

1,102,000 

151 

Tension. 

TRANSVERSE   STRENGTH  OF    TIMBER. 


679 


SPRUCE    BEAMS. 


Q. 

§ 

bit 

.5 
•a 

££ 
|| 

!^ 

4)         C 

& 

1 

1 

1 

_  4> 

^1 

£'S£ 

w 

"§ 

^  o 

0 

bo 

"So. 
1/5  c/r 

en  c  U 

?M 

Manner  of  Breaking. 

0 

6 

rt 

.<2</5 

u 
C 

c 
ed 

1 

2."*  rt 
3        3 

^  {/^  i/ 
3^  v 

•3  o  o 

* 

Q 

X 

M 

s""" 

* 

U 

inches. 

ft.    in. 

Ibs. 

398 

3^X12 

17      o 

Centre 

10,677 

5,854 

.602,000 

T74 

Tension. 

400 

2ifx,ofr 

15         0 

" 

5,725 

5,361 

,433,400 

152 

" 

401 

" 

4,107 

6,221 

,609,600 

170 

" 

404 

3    x  10 

15       6 

" 

6,828 

6.350 

,563,900 

173 

" 

4°5 
407 

4    x  12 

'5      o 
16      o 

" 

7,160 
8,660 

3,356 
4,287 

,244.300 
,099,300 

"3 

Shearing. 
Tension. 

408 

4     x  I2f*8 

16      o 

" 

8,762 

4,248 

,280,300 

136 

" 

4OC 

3*jj  x  12 

15       6 

44 

6,152 

1    127 

986,000 

102 

411 
414 

4X5 

4x12 

2^X103 

17      6 
16      o 

12  points 
Centre 

9,!57 

5,200 

3,700 

2,645 
4.902 
3,723 

,117,000 

,270,500 
,105,700 

143 

'  Tension. 
Crushing  at  knot. 
Tension. 

416 

2&x9ri 

15      6 

" 

7,900 

8,087 

,548,400 

215 

Longitudinal  shear  and  ten- 

sion. 

417 
421 

£"! 

14      o 

15         0 

u 

12,100 
I2.OOO 

5,139 

6,010 

1,622,500 
1,318,000 

182 
199 

Tension. 
Tension     and    longitudinal 

sdlear. 

422 

3lx  i  :f 

14      o 

" 

13,200 

6,182 

1,304,700 

217 

Tension. 

423 

4x12 

14      o 

" 

13,260 

5,762 

1,213,100 

207 

Tension,      crushing,      anc 

longitudinal  shear. 

424 

3^.  x  nf 

16      o 

" 

I2,OOO 

7,105 

2,100,000 

221 

Tension. 

425 

4     XI2 

14      o 

" 

9,710 

4,206 

1,067,800 

152 

" 

426 

4^  x  12 

14      o 

" 

6,l6o 

2,562 

966,880 

93 

" 

427 

4    x  I2± 

14      o 

" 

8,110 

3,663 

1,249,900 

135 

ii 

428 

429 

4    x  12 

14      o 

*' 

8,665 

3-753 

1,148,300 

135 

ii 

43° 

2f  X  10 

14      o 

" 

4,440 

4,217 

1,377,600 

127 

" 

43  i 

2jx  10 

14       o 

" 

6,500 

5,897 

1,464,200 

177 

it 

432 

2}XIO 

14      o 

" 

5,5oo 

4-991 

1,156,400 

Ii 

433 

ziixioA, 

14      o 

" 

2,925 

2.542 

896,800 

'78 

" 

434 

2*1X10 

14      o 

ii 

7,020 

6,525 

1,819,900 

196 

(i 

435 

•2\  X   IO 

14       o 

** 

4,990 

1,180,300 

136 

II 

436 

•2\  X  9^| 

14       o 

" 

5,120 

4^830 

1,242.800 

139 

II 

438 

2  J  X  9$ 

14      o 

" 

6,400 

5-969 

1,358,100 

177 

" 

439 

2jx  10 

14      o 

4i 

6,000 

5,467 

1,263.700 

164 

II 

441 

2^X  10 

14      o 

" 

8,080 

7,347 

1,732,800 

220 

" 

442 

i  \\  x  gT9fl 

II         0 

ii 

4,T3° 

4-5'9 

1,368,000 

1  66 

II 

443 

4    x  12 

14      o 

" 

0,760 

4,780 

1,  129,800 

172 

" 

444 
445 

4     X  12 

14      o 
14      o 

M 

0,000 
7,875 

4,495 
3,673 

1,238,200   ; 
1,327,600 

163 
133 

Longitudinal  shear  and  ten- 

I 

sion. 

446 

4     XI2 

14      o 

" 

0,400 

4,633 

1,286,800 

167 

Tension. 

44S 

14      o 

u 

0,800 

4,639 

I   236,400 

167 

" 

449 
45° 

3f  XI2 

14      o 
14      o 

II 

8,500 
0,400 

3,929 
4,658 

1,213,000 
1,160,900 

142 
166 

Longitudinal  shear  and  ten- 

sion. 

456 

1\\  x  Iaf 

19      o 

u 

4,000 

4,098 

1.267,000 

"3 

Tension. 

457 

7^  X  12 

19      o 

1C 

4,750 

4,456 

1,176,700 

121 

44 

458 

7*XI2 

19      o 

II 

6,95° 

5,216 

1,440,000 

139 

(4 

680 


APPLJED    MECHANICS 


SPRUCE   BEAMS. 


c 

bib 

53 

sS 

•sl 

£s.d 

JS 

I 

B 

u 

•3 

CO 

•o 
1 

II 

»$i 

|c| 

1 

•o 
rt 

a 

§& 

0 

•J 
be 

c 

11. 

s-sg 

ill 

Manner  of  Breaking. 

s^ 

•5 

rt  s 

c 

^2 

*j^    >  ,C 

3       3 

*^ 

o 

."2 

II 

c 

rt 

g 

•gee 

•a  >>cr 

0.-V3 

•3*0  s. 

^ 

Q 

s 

« 

s"" 

S 

U 

inches. 

ft.    in. 

459 

6  X  12 

14      o 

Centre 

11,600 

3,345 

915,000 

123 

Tension. 

460 
461 

6     X  12 

14      o 
14      o 

16,350 
15,600 

7,58o 
4»59° 

,640,800 
,369,000 

184 
166 

Crushing,  tension.and  shear 
Shear. 

481 

5i  x  *2 

15      o 

21,000 

6,120 

,845,000 

231 

" 

488 
489 

6}XI2 

6   xnj 

5fXI2 

19       o 

22         0 

4      o 

11,700 
13,500 

4,900 
5,8oo 
4.070 

,271,000 
,549,000 
,117,000 

130 
129 

Tension. 
Crushing  and  tension. 
Tension. 

495 

5f  x  12 

15,500 

4,660 

,195,000 

1  68 

u 

496 

6X12 

0         0 

22,500 

4,700 

,186,000 

236 

Shear. 

497 

6     X  12 

o      o 

24,600 

5,150 

258 

*fc 

498 

6     X  12 

5,55° 

1,850 

923,000 

60 

Tension. 

499 

6     X  12 

5       ° 

16,700 

5,230 

1,274,000 

176 

44 

501 

6     X12 

6      o 

10,800 

3,640 

948,000 

"5 

Average  vs 

lues 

4,52i 

1,310,584 

MAPLE    BEAMS. 


1 

;h  and  Depth. 

ince  between 
pports. 

ner  of  Loading. 

•o 
g 

be 
_c 

ulus  of  Rupture 
Lbs.  per  Square 
:h. 

ulus  of  Elastic- 
in  Lbs.  per 
uare  Inch. 

isity  of  Shear 
Lbs.  per  Square 
:h. 

Manner  of 
Breaking. 

o 

3 

13 

c 

rt 

? 

l'£~ 

T3  XCT 

o.-w 

|.s\s 

* 

Q 

s 

pq 

S 

^ 

*•» 

inches. 

ft.    in. 

463 
467 

4     XI2 

4    x  12 

XI      6 

10         0 

Centre 

9,650 
12,300 

6,200 

*   ,396,a«D 
,627,000 

III 

Tension. 

469 

4    x  12 

13      o 

44 

17,800 

7,280 

,448,000 

282 

*• 

47° 

14      o 

" 

9,200 

4,260 

,262,000 

T5° 

c« 

473 

3i  x  12 

15         0 

44 

14,600 

7.900 

,587,000 

265 

" 

475 

3£*"i 

14      o 

44 

1  8,450 

8,570 

1,597,000 

305 

<i 

477 

14       o 

15,500 

1  1,  080 

,702,000 

385 

Average  vah 

les  

7,146 

1,517,000 

TRANSVERSE   STRENGTH  OF   TIMBER. 


68  I 


Yellow-Pine  Beams. — The  moduli   of  rupture  in  common 
use  are  given  as  follows  by  different  authorities ;  viz.,  — 


Maximum. 

M  i  ni  mil  m  . 

Mean. 

Hatfield     .... 

2II68 

9000 

IS300 

Laslett  , 

14162 

10044 

12254 

C  Yellow  pine 

9000 

Trautwine  .... 

_ 

/ 

(  Pitch  pine 

9900 

Rodman     .... 

9876 

8796 

9293 

A  summary  of  the  figures  obtained  from  these  tests  will  be 
given  in  a  table  at  the  end  of  these  remarks. 
It  will  be  observed  that  we  have  for 


Maximum. 

Minimum. 

Mean. 

Modulus  of  rupture  .     . 

II360 

3963 

7486 

Modulus  of  elasticity     . 

2386096 

1162467 

1757900 

If  by  means  of  the  ordinary  deflection  formulae  we  wish  to 
compute  the  deflection  which  a  yellow-pine  beam  will  acquire 
under  a  given  load  after  it  has  been  applied  for  a  long  time, 
vve  should  use  for  modulus  of  elasticity  in  the  formulae  about 
one-half  of  the  values  above,  or  about  878950  (see  report  of 
time  tests). 

For  the  modulus  of  rupture  of  yellow  pine  of  fair  quality, 
in  the  light  of  the  tables  on  pages  683  and  684,  I  should  not 
feel  justified  in  using  a  number  greater  than  5000  pounds  per 
square  inch.  With  a  factor  of  safety  four  we  should  have 
about  1200  as  our  greatest  allowable  fibre-stress. 


682 


APPLIED    MECHANICS 


YELLOW-PINE    BEAMS. 


J 

C 

8 

i 

§•8. 

•s  k 

fj'-g* 

Q. 
& 

« 

1 

1 

Jj§-5 

|.s| 

1 

! 

S 

0 

J 

oJ£ 

"S-J^S 

•°  rt  * 

Manner  of  Breaking 

H 

rt 

°  Q, 

tl 

c 

—  ""  c3 

3.S  "~ 

rt^  cr 

*o 

5 

rt  a 

is 

3  J*  3 

3         D 

*2  *S) 

"O 

«-•  3 

c  5P 

s 

*O  3   O* 

*T3   ^»  O* 

U         ^" 

6 

S 

5" 

fl 

£ 

m 

!~c 

l'"c 

g-oS. 

inches. 

ft.    in. 

Ihs. 

3° 
32 

3   x  i34 

14      o 
18      o 

Centre 

15,158 
13,751 

6,614 

7,383 

1,937,000 
1,734,000 



Shearing.          [shearing. 
Tension,  crushing,   and 

33 

3ilx    2\ 

18      o 

** 

9,832 

5,^86 

I   7O4  OOO 

Shearing. 

47 

3    x    3J- 

14      o 

4< 

19,574 

8,696 

2'386,ooo 

359 

Crushing. 

50 
53 

4    x    4^ 

21          0 

24      6 

44 

12,875 

10,076 

57,lo6 

1,256,300 
1,784,400 

179 

Shtaring.          [shearing. 
Tension,   crushing,  and; 

54 

3    x    24 

24      o 

44 

9,576 

9,380 

2,116,800 

203 

Crushing. 

56 
57 

Six    4 

2*1  x     2 

J5       4 
19       2 

44 

io,572 
8,472 

4,764 
6,95° 

1,490,400 
1,444,500 

185 
182 

Tension. 

59 
62 

9     X      3f 

44  x  24 

24      o 
19     10 

" 

21,083 
15,461 

5,352 
9,102 

1,417,800 
2,038,000 

133 
237 

Tension  and  crushing. 
Crushing. 

63 

4i3Bx    2f»6 

20      o 

44 

14,073 

8,145 

1.599,000 

211 

Tension. 

64 

19     10 

41 

io,573 

6,098 

1,918,000 

161 

•' 

4    x    2} 

19      8 

44 

n,573 

6,782 

1,966,700 

183 

Crushing. 

67 

18      6 

44 

13,374 

7,277 

1,787,600 

196 

Tension. 

68 

4    x    24 

19      9 

44 

17,676 

10,872 

2,381,700 

278 

•' 

69 

3  IB  x    4 

20         0 

41 

6,675 

3,963 

1,169,300 

US 

Crushing. 

4i  x    2 

18      2 

44 

16,074 

8,248 

1,512,200 

227 

Tension. 

74 

4x2 

20         0 

44 

11,071 

7,004 

1,628,100 

175 

" 

75 

4    x    if 

19       9 

44 

I3,77i 

9,39T 

1,850,700 

233 

44 

76 

J7       4 

12  points 

15,825 

4,207 

1,344,100 

233 

44 

77 

4i  X      2 

17       4 

41            44 

37,325 

10,286 

2,123,200 

55  1 

11 

78 

4   x    24 

22       IO 

Centre 

7,172 

4,845 

1,455,300 

109 



79 

4x2 

19      8 

It 

2,087,600 

81 
82 

4x2 

11      8 

12  points 
Centre 

16,025 
I5,57i 

4,349 
9,671 

1,162,500 
1,607,300 

238 
246 

Tension. 

84 

4-i"  *  2i 

21       4 

44 

6.985 

1,501  .000 

167 

85 

4    x    if 

20         6 

4' 

lo'J™ 

11,360 

2,246,200 

1U/ 

271 

Tension. 

87 

4x21 

21       4 

44 

11,272 

7,335 

i,535,6oo 

175 

u 

88 

6      X      2;J 

20       4 

14 

15,283 

6,112 

1,613,000 

161 

t( 

9' 

4X2 

19     10 

44 

18,074 

2,223,800 

282 

Tension  and  shearing. 

92 

6x2 

6      5 

144 

21          0 

tl 

9,433 

6,012 

1,628,100 

MS 

Tension. 

145 

44  x    24 

19      3 

** 

11,201 

6,400 

1,472,000 

191 

" 

146 

4x2 

19      4 

" 

13,34! 

8,060 

1,839.700 

206 

" 

216 
218 
219 

4x2 

4&X      2j 

19      2 

:55  I 

iS      6 

44 
44 

16,748 
15,453 

16,632 

10,032 

7,040 
7,484 
7,427 

2,286,000 

2,010,300 
1,623,800 

244 

244 

Shearing. 
Crushing  and  shearing. 
Crushing  and  tension. 
Tension. 

220 

44  x    2^ 

iS      6 

44 

17,710 

7,982 

1,775,100 

265 

" 

221 

44  x    2^3 

16      o 

44 

16,330 

7,836 

1,931.200 

248 

k4 

222 

44  x    2^ 

16      o 

44 

18,515 

8,884 

1,786,000 

285 

44 

223 

4x2 

17      o 

44 

13,492 

7,167 

1,638,500 

212 

Crushing. 

224 

3il  x    i^l 

17      o 

44 

16,426 

8,958 

1.938,900 

264 

Tension. 

225 

4ix    2 

iS      6 

4( 

18,705 

8,786 

1,729,900 

285 

41 

226 

44  x    2 

iS      6 

44 

16,594 

7,794 

1,611,200 

253 

4i 

252 

3^f  x    i  If 

17      o 

44 

1  1,  006 

6,002 

1,271,000 

I77 

44 

254 

4x2 

16      o 

44 

15,226 

7,613 

1,617,700 

240 

41 

255 

4AX     2f 

iS      6 

44 

19,425 

7,975 

2,214,400 

267 

Crushing. 

257 

4ix   24 

16      o 

44 

17,424 

8,031 

1,789,800 

256 

44 

258 

4fgX     2 

16      o 

44 

18,319 

8,256 

1,830,000 

200 

Tension. 

26l 

418  X     2 

iS      o 

44 

17,818 

7,978 

1,989,700 

268 

44 

324 

44 

12,510 

6,002 

1,719,500 

206 

Shearing. 

326 

4    x    24 

16      o 

44 

14,011 

6,862 

1,980,600 

219 

Tension. 

329 

4i  x    a£ 

IS         0 

14 

23,324 

9,97i 

1,845,600 

34° 

44 

TRANSVERSE   STRENGTH  OF  TIMBER. 


683 


YELLOW-PINE    BEAMS. 


I     a 

hi) 

4>  1) 

,  u 

>•«• 

•5       i    8 

c 

3    3 

'-     Q_ 

'35  "j  •? 

o> 

I 

•5 

•d      i  3" 

rt 

£*"  c 

Q 

1u 

jj 

S            u 

3^'| 

4J     C  ~^ 

QJ 

•a 

2 

Q 

-1          o  o. 

"o    *-" 

•o  SS  5 

Manner  of  Breaking. 

H 

43 

II 

U 

bO                     •*:    y}' 

3         "B  J-g 

1-sS 

3         = 

P? 

0- 

3 

Is 

c 

o.Si5 

•0  >>CT 

o  •*-*  c/) 

U«*H     «J 

"x  °  a 

2 

* 

Q 

^ 

«        S  " 

* 

U 

inches. 

ft.   in. 

Ibs. 

331 

3JX  nl 

15       o 

Centre 

13,709       6.774 

1,619,800 

225 

Tension. 

33* 

4    x  I3i 

'5       ° 

41 

21,733   :      8.356 

2.109,600 

310 

Shearing. 

334 

*k 

17.400    :        7.105 

1,586,600 

259 

Tension. 

335 

4    x  12 

T5       o 

" 

16,228        7,686 

1,91  1.400 

257 

Shearing. 

339 

16       6 

" 

^,1M         8,373 

2,128,800 

250 

Tension. 

342 

4  3  x  I2i 

16       o 

" 

9,616   i      4.502 

1,380,500 

142 

346       4    xn< 

16       o 

'« 

16,227   i      8.284 

1,857,800 

256 

" 

347j      4    x  12 

17       o 

•' 

13.536 

7.I91 

2,041,000 

211 

350!        4      X    T2 

1  r          o 

4k 

10,212 

8,0'2 

1.790.500       299 

fcfc 

352!      4^x12$- 

17       o 

\i       o 

u 

18,238             9,087 
22.546          10.278 

2,059  ooo 
2.595,600 

27I 
331 

Shearing. 
Tension  &  compression. 

358       4ixi-4 

17       o 

41 

20.825     !     10.371          2.137.340   i        312 

"            " 

360 

3    x  10 

"   . 

12,  0,  I     '     10,864 

2,122,000 

302 

Tension. 

3    x  10 

T5       ° 

" 

7  264        6,485 

1.632,700 

179 

36  -> 

3    xgj 

(1 

8,467 

7,815 

1,833,700        214 

Shearing. 

364 

4    x  i2± 

15       o 

" 

15,208       6,879   i  1,667,500       237 

Tension. 

368 

15       o 

" 

17,714  1    8,133  i  1,899,100      276 

" 

376 
381 

3i  x  ITI 

15       ° 
J5       o 

!! 

14,712      7,388 

'5,201         7.761 

1,800,300       246 
1,566,400       258 

Tension  and  shearing. 

382 

4    x  i2± 

15       o 

" 

17,525  i    7,803 

1,882,700 

2(29 

Compression. 

386 

4    x  12 

16       o 

44 

12.110             6,055 

1,310,000 

191 

Tension. 

388 

4    x  12^ 

16      o 

" 

10,534   i     5,059 

1.658.800 

Shearing  and  tension. 

4      x   12^ 

'5       ° 

44 

22,100       10.147 

2,234,800       342 

Shearing. 

39  1 

4&  x  12 

17       o 

44 

18,391    1     9,474 

2,068,200  I     279 

Tension. 

394 

4i  x  I,?* 

16       o 

" 

19,713   |     8.902 

1,096,400 

286 

Shearing. 

396 

3^x12 

20       o 

" 

10,296        7.354 

1,889.000 

186 

Tension. 

8       6 

u 

25,827        7,392 

2,217,000 

445 

Shearing. 

399 

4ix  12^ 

16      o 

" 

8,602 

3,828 

1,306,000 

124 

Tension. 

402 

4f  X  12^ 

15      o 

" 

15.192 

6.303 

1,542,000 

215 

*fc 

403 

4i  X  12J- 

'5      o 

" 

10,886 

4.636 

1,058,200 

158 

" 

410 

3JX12f 

15         0 

2  points  2' 
from  cen. 

18,590 

6.304 

1.733.400 

297 

" 

412 

4£X  12^ 

14      o 

do.     do. 

15.865 

4,627 

1,774,100 

237 

Shearing. 

413 

3^  x  12$- 

'5       o 

do.    do. 

14,852 

5.069 

1,778,900 

236 

Tension. 

418 

4    x  12 

14       o 

Centre 

20,500 

9.064 

2,278,500 

326 

" 

4'9 

4    x  12 

16      o 

" 

12,850 

6,54i 

1,664,800 

207 

Crushing  at  top  &  shear- 

ing above  neutral  axis. 

420 

4    x  12 

14      6 

" 

14,000 

6.442 

1,684,400 

225 

Tension. 

429 

4    x  12 

14      o 

11 

15,600 

6,759 

1,498,500 

244 

Tension,  compression.  & 

longitudinal  shear. 

437 

6    x  i6£ 

17      6 

*« 

38,000 

7.407 

1,777,700 

289 

Tension  and  shear. 

440 

4x12 

16       o 

" 

19,500 

9.908 

1,778,400 

3'2 

Tension. 

447 

3$  x  12^ 

18       o 

Vk 

13,900        7,908 

1,672,900 

227 

'• 

451 

4      X  I2\ 

18       6 

" 

9-  3  .so 

S-446 

2.013,200        151 

Longitudinal  shear  and 

tension. 

45? 

4^x  I2j% 

14      o 

it 

I5i75o 

6,57° 

1,704,900       240 

Tension   and    longitudi- 

nal shear. 

453 

4     X12 

14      o 

" 

13,000 

5.800 

1,804,000       209 

Tension 

454 

4i  X   12 

10         0 

" 

12,000 

6.848 

1,603.800        182 

Tension  &  compression. 

435      4^x12        16       o 

16,750 

8.200 

1.656,000       259 

Tension. 

Average  values.  ... 

7,442  j  1.783,000 

684 


APPLIED   MECHANICS. 


•si 

S  J  .S 

M      l/l 

^            T3 

o    £ 

u  -S    ^ 

•*  * 

1  '«  s 

E  g 

B  *5  .. 

y  oo 

3     rt    ^ 

•S 

i 

»J    -1 

£ 

Ml 

?!~!      s 

-  o   ? 

-•as-           | 

•  .c    ^ 

^  3  *M  &.              ° 

Q    ^J     in 

Q      CJ      !^      C/l                                 rj 

z                      o 

^                         A 

•g  ^ 

O^cooo     O    OOcooo 

£'0    rfcoooo     Tj-r^oo    0     o 

<«  .ti 
11 

t'rf   \n   o    w    N     to    u-> 
N     OO      CO   CO      N      M      C 

rVo   O    OO»coo    w    MOOCO 

l« 

2  si 

00,^000^ 

^•XOTJ-CO^O      VOOCO      CO     "f 
C>O      t^O      CMCO      fiCO      COM      N 

3  a 

xn  \o    ^f   r**   w>   xo   co   *o 

VO     ^   CO      XO     ^*   vO      CO    U">    vO      ^"     VO 

°<5 

S* 

C   £    uj 
13    h/D^ 

\O      vO       M       M       M       ^     M       CO 

r**»    o    ^O    co    ^  oo    co    t^* 
coa^O    coco    r->«co 

-tvocoo     O     O    CON     O^r-Q 

O    CO      O^    ^   CO    vO      vo  CO     vO    CO      O 
wcoOvoOcoo^ONdO 
vo   r^*    O     O   co    CM    vo    o     O   vO    t^*1 

££•9 

M        M        M        M                    M 

M                    CM       M                    M                    M        t-l 

03 

"5 

c 
o 

"O 

a 

E 

•o 

° 

J 

a     ovocoo-^-covoo 

rtvOOOOOOOOOvO 

a 

O^vncoco    mmino^ 

ONC^O     I^OvvO     0><^C>vot^ 

- 

•a 

r*»     ^vf1  *,££    f^    *!)tf  *!i?     «x    M 

•ff  .,    e,    «,    „    „    „    „    «  'Tff 

rt  .£ 

•g   xxxxxxxx 

xxxxxxxxxxx 

B 

<-f« 

vO     Tfto^trfO     co^i- 

ce(ao                                                                     *•*» 

"o  ~ 

oo*HvnOO^OMN 

co^O^coOvaco^OO 

IH 

TRANSVERSE   STRENGTH  OF   TIMBER. 


68; 


-o  s 


3  J3  I 
o  »,= 

'S  S  8 
S  "  * 
1.2  •£• 

'".Crt 

lit 


*     ll 

!K!l 


CB    .ti 

9    u 


R88888888R8888888    8 

oo  \O  0^  w>  p*  d  \o  u*>  co  ^  co  co  ^f  O  O  w  co 

O   *O  CO  ^"  CO   "^  ^"  U^  CO   t"**  OO   W  OO   O  CO   d   M    OO 

^"  vO   vO    r***   \ti  10   ^    ir>  t^   u^  co   sO    ^*  O   ^O   O    r*» 

M       W       d       '<fl/>C*       M       M       TfM       COO^OO       ^OOOOO 

co   co   w    r**   Is**   I****    O    '"^   f^   co    ^  vO    co    ^  d   co    oo          • 

^ 

1,5!!:!!:::5:3:!!  : 

I   xxxxxxxxxxxxxxxxx 

HOOIHOO  o»|«Nao  ^^  H»'<HrV  H* 


ii 
11 


J'C  "^ 
&-s 


I 


1,- 


M    N    cnco     O"-"    N    c^^tnvO     OOW    tir>c> 
xr>   in   \n  O    O    r^f^r^r^r^f^moOOOO 


686 


APPLIED    MECHANICS. 


WHITE-PINE   BEAMS. 


No.  of 
Test. 

Width  and 
Depth. 

Span. 

Breaking 
Centre 
Load,  in 
Ibs. 

Modulus  of 
Rupture. 

Modulus  of 
Elasticity. 

Remarks. 

inches. 

ft.  in. 

94 

3   X  Hi 

15  8 

5088 

3613 

924252 

(  Pattern  stock. 

95 

3   X  13 

14  o 

12588 

7251 

1280832 

•j  Clear  piece. 

(  Seasoned  3  yrs. 

96 

.3   X  13 

16  6 

9088 

5324 

1072889 

97 

3   X  ii 

15  8 

6088 

4729 

978256 

98 

2j  X  9f 

16  o 

6088 

-6415 

1234880 

99 

2f  X  13 

15  6 

5988 

3438 

1020390 

IOO 

3   X  9f 

16  o 

4288 

4330 

H65937 

102 

3   X  io| 

15  6 

4790 

3855 

999190 

103 

3   X  ii 

16  6 

6588 

5390 

1242649 

104 

3   X  ni 

15  6 

5088 

3739 

931760 

128 

6   X  12 

19  10 

12922 

5340 

1380660 

129 

6  X  12 

19  08 

15060 

6170 

1565000 

130 

6  X  i2i 

19  08 

12340 

4954 

I222IOO 

131 

6  X  12 

19  10 

13023 

538o 

1307900 

132 

6   X  12 

19  10 

6231 

2575 

II03500 

133 

6   X  12 

19  08 

12912 

5290 

I297OOO 

134 

6   X  12 

20  oo 

11254 

4689 

1345700 

137 

6&  X  i*fc 

19  08 

13650 

5478 

1367700 

138 

6  X  12 

19  09 

14010 

5765 

I247IOO 

140 

6   X  12 

19  °9 

9761 

4016 

IIO56OO 

244 

4A  X  iaH 

16  oo 

10179 

4300 

948600 

245 

4i  X  "i 

15  6 

12984 

5620 

I27IOOO 

247 

4T*  X  io| 

15  8 

6770 

3638 

IIII9OO 

248 

4i  X  12* 

15  8 

7790 

3549 

1057300 

279 

4  X  i2i 

15  6 

9085 

4222 

1084000 

280 

4   X  12* 

15  6 

7575 

3521 

854300 

28l 

3*  X  12^ 

15  o 

12070 

5547 

I28IOOO 

282 

3tt  X  I2TV 

15  o 

8660 

3998 

1053000 

283 

3if  X  I2i 

15   0 

7182 

3285 

970300 

284 

4   X  i2i 

15  o 

5685 

2456 

873300 

285 

3tt  X  I2A 

15  6 

6385 

3031 

901500 

286 

3i  X  12  A 

15  6 

11791 

5449 

1258700 

287 

4   X  I2& 

15  6 

8265 

3802 

1049500 

288 

31  X  i2T5s 

15  6 

11272 

5353 

1295382 

289 

31  X  12  A 

15  6 

5671 

2806 

825300 

296 

4   X  I2& 

15  6 

557i 

2563 

7272OO 

315 

31  X  12 

16  o 

7165 

3820 

H584II. 

Average  valu< 

4451 

II22OOO 

It  would  seem  to  the  writer  that  about  the  same  modulus 
of  rupture  should  be  used  for  white  pine  as  for  spruce. 


TRANSVERSE   STRENGTH  OF    TIMBER. 


687 


KILN-DRIED   WESTERN   WHITE   PINE. 


No.  of 
Test. 

Depth  and 
Width. 

Span. 

Manner 
of 
Loading. 

Breaking 
Load  (Ibs). 

Modulus  of 
Rupture 
(Ibs.  per 
Sq.  In.). 

Modulus  of 
Elasticity 
(Ibs.  per 
Sq.  In.). 

Remarks. 

inches. 

ft.    in. 

206 

2TV  X  13! 

15   05 

Centre 

9325 

7014 

1505000 

207 

2        X    I2| 

13  oo 

8814 

6432 

II93000 

208 

2TV  X   I2T9* 

15  oo 

3420 

2836 

752300 

2IO 

2TV  X  Hf 

15  oo 

4518 

4284 

1241400 

212 
213 

2TV  X   13 

2TV  X  i3rf 

15  06 
15  06 

6T2O 

4898 

1099300 
1276300 

214 

2        X   I2TV 

15  06 

7420 

6430 

II66OOO 

237 

23     NX    1  2*4 
W%  S\  *•  ^Te 

15  04 

8225 

6478 

I23IOOO 

Mean  = 

5482 

1183037 

HEMLOCK. 


.  of  Test.] 

Depth  and 
Width. 

Span. 

Manner 
of 
Loading. 

Breaking 
Load  (Ibs.). 

Modulus 
of 
Rupture 
(Ibs.  per 

Modulus  ol 
Elasticity 
(Ibs.  per 

Remarks. 

o 

Sq.  In.). 

Sq.  In  ). 

inches. 

ft.    in. 

154 

155 
156 

3i    X  n| 
3TV  X  10 
2|    X    9f 

15  oo 

14  08 
12   09 

Centre 

6449 
4648 
4425 

3965 
4007 
3716 

870960 
971710 
750400 

Nos.  154-160  are 
Eastern     hemlock 
seasoned  about    i 
year.       The       re- 

r57 

2T3g  x  io| 

II    10 

3223 

2381 

770900 

mainder     of      the 

.'53 

159 
1  60 

3  A  X  IOT\ 
3      X    9l 

2|     X   Hi 

14  oo 

13    06 
12    08 

4137 
2939 
9433 

3151 
2570 

5911 

735800 
833000 
1086600 

hemlock   tests  are 
from   hemlock  cut 
in  Vermont  June, 
1886;   first  growth 

177 

178 

4*    X  12 
4      X  12 

15    08 

17  oo 

9605 
5502 

4560 
2923 

1081100 
821990 

grown     on      high 
ground.    Sawed 
Sept.  25  1886.    Re- 

:79 

4|    X  iitf 

15  08 

3192 

688960 

ceived  at  Institute 

1  80 

4      X  i2£ 

15  06 

4584 

2175 

926560 

Nov.  15,  1886.  Test- 

181 
182 

4      X  nH 
4      X  12 

15  08 

15  06 

5133 
11073 

2539 
5363 

758390 
1296600 

ed  Dec.  2,  1886,  to 
March  9,  1887. 

183 

4      X  12 

15  06 

13274 

6499 

1269800 

184 

3l    X  u| 

15  08 

9964 

5142 

1075600 

185 

3i    Xn« 

17   02 

3679 

2059 

412670 

186 

3ff  X  ntf 

15    08 

' 

12488 

6535 

1327200 

Mean  = 

3825 

922250 

114 

71    X  ioi 

20  04 

Centre 

19244 

8243 

2011188 

Yellow  birch.N.H. 

1  20 

7i    X  io| 

19  C>6 

" 

16150 

7627 

1583201 

"           "        4t 

1 

3\/     TQ 

13   06 

,, 

1406? 

12122 

1  N.  H.  ash,  sea- 

f\     A  v 

**fV    3 

")  soned  2  yrs. 

688 


A  P  PLIED    ME  CffA  HICS. 


TIME    TESTS. 

The  following  is  a  record  of  the  time  tests  made  at  the 
Institute,  and  at  the  close  will  be  found  a  statement  in  regard 
to  the  proper  value  of  modulus  of  elasticity  for  use  in  com- 
puting deflections. 

TIME    TEST    NO.     I. 

Spruce  from  Maine,  received  at  Institute  October  30,  1885. 
All  the  beams  when  received  were  green  lumber,  except  F, 
which  was  seasoned  on  the  wharf  about  six  months.  Beams 
A,  B,  C,  D,  E,  and  F  were  seasoned  under  a  centre  load  in 
the  laboratory  in  steam  heat  from  November  10,  1885,  to  May 
8,  1886.  Beams  G,  H,  I  were  seasoned  in  the  same  room, 
without  load ;  span  =  20  feet  for  all  the  beams  under  load. 


Beam. 

A  (164) 

B  (163) 

C  (162) 

D  (161) 

E  (169) 

F  (168) 

Description  of  lumber,  . 
Dimensions]  -l^1'  ; 

(  with'wt.  of 
Max.  fibre         beam,     . 
stress,  Ibs.  -|  without 
per  sq.  in.         wt.     of 
beam,     . 
Deflection,  load  first  ap- 

clear 
4"  x  12" 
3i"xnf" 

1078 

1003 
o".S488 

i''.oi64 
1462200 
789000 

6574 

6500 
35-5 

27.1 
May  ii,  '86 

1866900 

knotty 

4"  X  12" 

4"  xi  if" 
1076 

1003 
o".63Ss 

i".47°7 
1262900 
546000 

7260 

7187 

34-9 

28.1 
May  1  1,  '86 

1509500 

knotty 
4"  x  12" 
3f"xii|" 

1074 

1003 
o".7675 

*"-S*54 
1045700 
529000 

5007 

4937 
33-6 

25.5 
May  10,  '86 

1367300 

clear 
4"  x  ia" 
3i"xnf" 

1070 

1003 
o".6io8 

1-3734 
1314000 
584000 

6066 

6000 

31-9 

25.7 

May  10,  '86 

1625500 

clear 

6"XI2" 

5|"xn|" 
"33 

1057 

o".7i88 

1.2667 
1175000 
667000 

6779 

6708 
34-1 

28.3 
May  14,  '86 

1737400 

clear 
6"  x  12" 
5l"xn|" 

1136 

1051 
o//.5454 

0.3151 
1540000 
2666000 

8574 

8500 
35-4 

2Q.6 

May  13,  '86 
1718300 

Deflection    at    end    of 
test,  
E  (immediate),      .    .    . 
E  (final)        .    . 

•Sfij&p 

sqin^    I    r*      °f 
I.   beam,    . 

Weight    per  cu.  ft.  at 
beginning,  Ibs.,      .     . 
Weight   per  cu.  ft.  at 
end,  Ibs.,  

Date  of  testing,    .    .    . 
E  (after  seasoning),  Ibs. 
per   sq.    in.    of   final 
section       .        .    .    • 

Average  E  (immediate)  =  1300000. 

Average  E  (final)  as    963500. 

Average  modulus  of  rupture  beams  under  load  •»  6710. 

All  quantities  in  the  above  table  except  the  last  were  calculated  by  Min?  the  origina. 


TRANSVERSE  Sl^RENGTH   OF   TIMBER. 


689 


Beam. 

G  (165) 

H(i66) 

Ki67) 

clear 

clear 

knotty 

4"  x  it" 

4"xia" 

4"  K  12  •*' 

Dimensions  {final                                  

3i"xiif" 

T±"  X  12" 

A"*I2' 

Modulus  of  rupture,  i  with  wt.  of  beam,  .    .    . 
Ibs.  per  sq.  it).           <  without  wt.  of  beam,  .    . 

65*5 

7187 

joia 

37 

May  ii,  '86 

May  12,  '86 

May  ia  '8f 

1  E  (af  ier  seasoning),  Ibs.  per  sq.  in.  of  final  section, 

1603700 

1748900 

1457000 

Average  E  (final  section),  beams  without  load, 
Average  modulus  of  rupture  (original  section). 


1603200 

6508 


TIME  TEST  FOR  SHORT  PERIODS  OF  TIME. 


Total  in- 

No. 

^  Depth  and 
Width. 

Span. 

Original 
Modulus  of 
Elasticity. 

creased 
Deflec- 
tion. 

Modulus 
of 
Rupture. 

Description  of  Test. 

inches. 

ft.  in. 

inches. 

Ibs. 

1  08 

6  X  12 

17    04 

1269670 

.1241 

5066 

Spruce-beam  load  equally  dis- 

tributed at  12  points.     The  , 

beam  was    subjected    to   a 

load  of  5031  Ibs.  for  898  hrs. 

148 

4i%  X  iitf 

17    04 

1211800 

•3765 

4668 

Yellow-pine   beam,   load  dis- 

tributed    equally    over     12 

points.     The  beam  was  sub- 

jected to  a  load  of  6355  Ibs. 

for  29  days. 

TIME    TEST    NO.    2. 

Spruce  beams  cut  in  Maine  in  the  spring  of  1886.  Received 
at  Institute  September  13,  1886.  Beams  A,  B,  C,  D,  E,  and 
F  were  seasoned  under  a  centre  load,  in  the  laboratory,  in 
steam  heat,  from  September  15,  1886,  to  April  2,  1887  (2O° 
days).  Beams  G,  H,  I  were  seasoned  in  the  same  room,  but 
were  not  loaded.  All  the  beams  were  without  load  between 


690 


GPPL IED    MI  CHA  NICS. 


April  2  and  date  of  testing.     Span  =  18'  oo"  for  all  beams 
under  load. 


Beam. 

A  (194) 

B  (193) 

C  (190 

Dfoo) 

B(i95> 

F(i9a) 

Description  of  lum- 

I 

straight- 

straight. 

ber,    . 

clear 

clear   ] 

cross- 
grained 

fknotty-J 

grained, 
some  knots 

grained, 
some  knots 

Dimen-  j  original,  . 

4"XI2" 

4"XI2& 

3f"xiiJ" 

4A"x«f" 

6"    XM&" 

6i"   xia" 

sions  1  final,    .    . 

3tt"«"tt" 

3*Fx,,tf" 

3f"X!lA" 

4i"  xiiJt" 

5f"xii|" 

sjrxnH" 

f  with 

wt.of 

Max.  fibre 

beam, 

I02O 

MIX 

"36 

994 

1095 

»93 

stress,  Ibs. 

with- 

per sq.  in. 

out 

wt.  of 

beam, 

066 

956 

1075 

943 

1037 

104X 

Deflection,load  first 

applied, 

.    .    . 

o".3707 

of'.4»53 

°"-5333 

<*"-4794 

o".so»4 

o"'5747 

Deflection  at  end  of 

test 

o".9i8j 

o".8447 

i".3336 

l".O4<» 

t".oi7o 

1^.5050 

E  (immediate),  .    . 

1689000 

1449000 

I333000 

*     ***TOO 

1388000 

1337000 

1173000 

E  (final), 

683000 

730000 

577000 

591000 

655000 

448000 

Modulus 

with 
+*f 

of  rup- 
ture Ibs  • 

wt.  ot 
beam, 

10983 

7838 

3943 

5038 

7448 

5116 

without 

per  sq. 

wt.  of 

in. 

beam, 

10339 

7783 

3182 

4087 

739* 

SC64 

Weight  per  cu.  ft 

at  beginning,  Ibs., 

31.8 

3»-8 

35-4 

30.0 

34-* 

30.5 

Weight  per  cu.  ft. 

at  end,  Ibs.,  .    .    . 

38.6 

•6.1 

«9.« 

87.0 

«7-7 

rf.3 

Date  of  testing, 

Apr.  25,  »8; 

Apr.  19,  '87 

Apr.  5,  *87 

Apr.  4,  ty 

Apr^.'S? 

Apr.  8,^87 

E  (after  seasoning), 

Ibs.  per  s 

q.  in.  of 

final  section,    .    . 

3125500 

1853300 

1731600 

1517800 

166*100 

U94JO» 

Average  B  (immediate)  «•  1376500. 
Average  E  (final)  —   614000. 

Average  modulus  of  rupture  beams  under 


TRANSVERSE   STRENGTH   OF   TIMBER 


69! 


Beam. 

G(i88) 

H  (187) 

i  (i89) 

knotty 

clear 

knotty 

4"  x  12" 

A"  x  12" 

*"x  12" 

Dimensions  |^^    \\\\\\\\\ 

3f"xiif" 

3*S"xn*" 

3*"  xx  i*4w 

Modulus  of  rupture,  j  with  wt.  of  beam,  .    .    . 
Ibs.  per  sq.  in.          1  without  wt.  of  beam,  .    . 
Weight  per  cu  ft  at  beginning,  Ibs.,  ... 

5218 
5i67 

8667 
8598 

7cr    g 

4796 
4751 

24.9 

£Q.^ 

24  8 

Date  of  testing                      .         . 

Mch  21  '87 

Aor  i  '87 

Mch  16  *87 

£  (after  seasoning),  Ibs.  per  sq.  in.  of  final  section, 

1355300 

1914500 

1573600 

Average  E  (final  section),  beams  without  load  =  1614500. 
Average  modulus  of  rupture  (original  section)  =*       6227. 

TIME    TEST    NO.    3. 

Yellow-pine  beams  from  Georgia,  cut  in  season  of  1886. 
Received  at  Institute  September  13,  1887.  The  lumber  was 
purchased  in  sticks  of  double  length,  and  cut  in  two  for  testing. 
The  numbers  indicate  the  stick,  and  the  letter  "  B  "  the  butt, 
and  <4T  "  the  top  end  of  the  same.  Beams  i  T,  2  B,  3  B,  4  T, 
5  B,  5  T  were  seasoned  under  load,  the  remainder  being  sea- 
soned without  load.  Span  =  iS1'. 


692 


APPLIED    MECHANICS. 


Beam. 

•  T(a38) 

28(239) 

38(340) 

4T(W, 

,»(,«,> 

,T«H,> 

Description   of  j 
lumber,    .    .| 

clear, 
cross- 
grained 

clear, 
some  dry 
rot 

clear, 
straight- 
grained 

clear, 
cross- 
grained 

clear, 
straight- 
grained 

clear, 
straight- 
grained 

Dimen-  j  original 

i  «-     * 

4lV'  X  I3/j" 

4ii"xi2&" 

4i"XI2/a" 

3*r  x  i2&^ 

6&^xnir| 

Wt***"n 

sions  » 

nnai,     . 
with 

4        xnH 

*A        A 

3ii      »ii 

3it"xxxir 

6"x»A" 

e^xusr 

Max. 

wt.  of 

fibre 

beam, 

1358 

1259 

1381 

U35 

1626 

1509 

stress, 

with- 

Ibs. per 

out 

sq.  in. 

wt.  of 

beam, 

ia83 

"93 

1308 

1304 

1531 

x4»6 

Deflection,    load 

first  applied,     . 

o".53io 

o".S5i8 

o".47oo 

o".56i8 

o".443« 

o".4836 

Deflection  at  end 

of  test,     .    .    . 

o".98i7 

i".a387 

o".9434 

i".4506 

0//.8364 

o".8449 

E  (immediate),    . 

1536000 

X35XOOO 

1763000 

1545000 

3290000 

i  880000 

E  (final),     .    .    . 

832000 

OO9OOO] 

879000 

596000 

13x3000 

1081000 

Modu- 

with 

lllO    f\f 

wt.  of 

JUS  OI 

beam, 

6390 

6234 

9545 

6570 

10630 

10000 

rup- 

with- 

ture, 

out 

Ibs.  per 

wt,  of 

sq.  in. 

beam, 

6*13 

6163 

940> 

0495 

10540 

99*4 

Weight  per  cu.  ft. 

at     beginning, 

Ibs  

43-6 

40.6 

48.8 

43-  X 

50.9 

45-8 

Weight  per  cu.  ft. 

at  end,  Ibs.  .    . 

38.4 

36.8 

43-3 

39-0 

45-8 

41.4 

Date  of  testing,  . 

May  aa,  »88 

May  38,  '88 

May  28,  "88 

May  28,  *88 

May  31,  '88 

June  x,  TO 

E  (after  season- 

ing),   Ibs.    per 

sq.  in.  of  final 

section,    .    .    . 

1898000 

1651000 

3340000 

1803700 

•501400 

3103400 

Average  E  (immediate)  «=  1729000. 
Average  E  (final)  m   867000. 

Average  modulus  of  rupture  beams  under  load  m  Sen. 


TRANSVERSE   STRENGTH  OF    TIMBER. 


693 


B~ 

tB(«o) 

«T<«33) 

3T(«3S) 

4B  (Ml) 

Description  of  lumber,     .    .    .    . 

i       clear, 
•j    straight. 
'     grained 
4!"  x  isA" 

dry  rot 

4JL"  X  1*1" 

unsound 

4  A"  X  12" 

clear, 
straight* 
grained 

~S1//  x  12JL» 

Dimensions]  fiQ*    "^    ;    ;    ; 

A!"  xix*" 

4A"  X  I2A" 

A"XII!&" 

,1  3'/  x  ,2// 

f  with     wt.     of 
Modulus  of  rup-        bean* 
ture,   Ibs.   Perjwithoutwt;o; 

**'"'                [     beam,  .    .    . 
Weight  per  cu.  ft.  at  beginning, 

Ibs 

9417 
9330 

46  I 

3069 

3<»7 

5838 
5754 

99<50 

089» 

Weight  per  cu.  ft.  at  end,  Ibs., 
Date  of  testing  
E  (after  seasoning),  Ibs.  per  sq. 
in.  cf  final  section,     

41.4 

March  28,  '88 
2141400 

April  6,  '88 
1393000 

April  9,  '88 
1951000 

March  26,  '88 

1895600 

Average  E  (final  section),  beams  without  load  as  1820200. 
Average  modulus  of  rupture  (original  section)  *      9071. 


TIME    TEST    NO.    4. 

Spruce  from  Maine.  Green  lumber.  Received  at  Insti- 
tute September  11,  1888.  Beams  A,  B,  C,  D,  J,  K  were  sea- 
soned under  load  in  the  laboratory  in  steam  heat  from  Sep- 
tember 13,  1888,  to  March  25,  1889  (194  days).  Beams  E,  F, 
G,  and  H  were  seasoned  in  the  same  room,  but  not  loaded, 
All  the  beams  were  without  load  from  March  25,  1889,  to  the 
date  of  testing.  Span  =  18'  for  the  beams  while  under  load. 


694 


A  PPL  IF.  D    ME  CH A  NICS. 


Beam. 

A  (273) 

B  (274) 

C  (276) 

D  (278) 

J0»77) 

K^S, 

Description  of  lum- 

i   knotty, 

clear, 

clear, 

ber,  ,    . 

,    .    .    . 

ave.  stock 

ave.  stock 

-J  straight- 

straight- 

straight- 

'  grained 

grained 

grained 

Dimen-  (  original,    . 

4i"xiifi" 

4TVx«ii" 

4ft"  xi  if" 

4£//xn|" 

5$|"  xi  i$|'; 

5J"  x  12" 

sions  (  final,     .    . 

4"  xi  i*" 

sir*  «ft" 

4"xiif' 

3i!"xn^" 

5ft"  x  ii|" 

Sf"  x  ii&" 

with 

Max.  fibre 

wt.  of 

stress, 

beam, 

1422 

»44» 

1444 

1436 

1750 

1705 

Ibs.    per 

without 

sq.  in. 

wt.  of 

beam, 

1364 

1385 

1387 

1379 

1698 

164$ 

Deflection,  load  first 

applied, 

o".66g9 

o".77i6 

o".6oi8 

o"'737° 

o".8446 

o".741O 

Deflection  at  end  of 

te^t 

i"  .  3023 

t".8375 

tff'S3^9 

l".6Q8o 

l"  73  SO 

i".4284 

E  (immediate), 

1326000 

1169000 

1387000 

1224000 

1308000] 

1431000 

E  (final) 

682000 

491000 

597000 

532000] 

637000 

745000 

Modulus 

with 
wt.  of 

of     rup- 
ture, Ibs. 

beam, 
without 

7545 

7211 

5597  ^ 

6622 

6484 

7099 

per     sq. 

wt.   of 

in. 

.   beam, 

7487 

7«55 

554° 

6565 

6431 

7038 

Weight  per  cu.  ft.  at 

beginning,  Ibs.,    . 

34-3 

33-5 

33-0 

33-6 

31.5 

364 

Weight  per  cu.  ft.  at 

end,  Ibs., 

.    . 

27.3 

25*4 

26.8 

a6.9 

97-1 

28.6 

Date  of  testing,  .    , 

Mch.  26/89 

Apr.  3,  '89 

Apr.  9,  ^89 

Apr.  n,  ^89 

Apr.  10,  »89 

Apr.  8,^89 

E  (after  seasoning), 

Ibs.  per  sq.  in.  of 

fioal  section,    .    . 

1631700 

1405500 

1747900 

1503200 

1576000 

1722600 

Average  E  (immediate)  as  1307500. 
Average  E  (final),  =    614000. 

Average  modulus  of  rupture  beams  under  load  «  £760. 


TRANSVERSE   STRENGTH  OF    TIMBER. 


695 


Beam. 

E(27o) 

F<871> 

G(27») 

H  (251) 

nearly 
clear, 

knotty 

straight- 
grained 
3!"  x  ia" 

4"xiiW" 

3§4"x  12" 

A*"  xi  i*" 

Dimensions  \&^  ....... 

3f"xii$" 

3lf"xii$?/ 

o  AS"  Xii  A" 

4tC"xnf" 

M^ulus  of  rupture,  ("j^-"^-^ 
]bs.  per  sq.  m.         |     beam            ^    t 

8293 

76i5 

8558 

Riofi 

4732 

Afllfi 

Weight  per  cu.  ft.  at  beginning,  Ibs.,    .    . 
Weight  per  cu.  ft.  at  end,  Ibs.,      .... 

33-5 
27.3 
Mch.  n  '80 

37-2 
27.6 
Mch   19  '89 

36.8 
26.3 

Mch  22  '89 

33-o 
Nov767  '88 

E  (after  seasoning),  Ibs.  per  sq.  in.  of  final 

1366500 

Average  E  (final  section),  beams  without  load 
Average  modulus  of  rupture  (original  section) 


1610000. 


DEFLECTIONS    WITH    TIME. 

From  the  above  it  is  plain  that  the  deflection  of  a  timber 
beam  under  a  long-continued  application  of  the  load  may  be 
2  or  more  times  that  assumed  when  the  load  was  first  applied  ; 
and  in  order  to  compute  it  by  means  of  the  ordinary  deflection 
formulae,  we  should  use  for  E  not  more  than  £  the  value  de- 
rived from  quick  tests. 

LONGITUDINAL    SHEARING. 

Below  are  given  tables  showing  the  greatest  intensity  of 
the  shear  at  the  neutral  axis  of  each  beam  at  fracture  as  calcu- 
lated from  the  formula  on  page  675. 

For  breaking  shearing-strength  per  square  inch,  in  the  case 
of  each  wood,  it  seems  to  the  author  that  it  would  be  proper 
to  use  a  value  somewhere  near  the  lowest  of  those  given  in 
the  table  of  beams  which  gave  way  by  longitudinal  shearing;. 

It  will  also  be  observed  that  these  shearing-forces  are  less 
than  those  obtained  from  the  experiments  on  direct  shearing 
along  the  grain,  made  at  the  Watertown  Arsenal  ;  and  this  is 
naturally  to  be  expected,  for  the  shearing  in  their  case  took 
place  along  a  section  that  was  perfectly  sound,  while  in  these 
cases  it  took  place  at  the  weakest  point. 


696 


APPLIED    MECHANICS. 


l 

.2^,  o* 


wig 

H  -S| 

s^i 


£  a 

c  8 


( 

fe 

o 

0                    0 

^ 

o 

W 

S|l 

XX 

0 
0 

1 

J 

8 

O                  to 

NO 

NO            <N 

? 

8 

o 
r^ 

"w  bo 

fe 

H 

« 

NO 

c 

•—  » 

ON 

O 

4l 

?  «»* 
XX 

o 

"<s 

^ 

o 

O                   00 

1      S 

NO      b 

"5        O 

o 

0> 

| 

o  ™  2 

M 

* 

o 

2" 

00 

to       to 

c 

f> 

SB 

H^z 

H 

4 

(M 

-ci1 

%% 

, 

^ 

o 
o 

0                    0 

O                   00 

1 

0 
O 

PH 

^*rt'§ 

XX 

H 

«*J 

1/3 

'•£ 

o 

fi       oo 

. 

M 

0  *^  SH 

v     ^ 

w 

. 

(^ 

NO 

ir>        ^ 

d 

rr) 

to  M 

S% 

M 

0] 

»—  I 

ts 

<*-!       | 

v     - 

f^ 

*      *       O 

H*HH 

0 

o             o 

O 

O 

wT^:  *2  5M  -0   • 

M     M 

o 

V 

^•N 

o 
o 

8       £ 

O         CM 

O 

3'1'i^^l 

XX 

Ov 

o 

£) 

NO                                       « 

oo*      fo 

: 

o 

to 

d 

tn  tiO  n  ^ 

"^Hoe 

M 

CT3 

N 

o   o   o 

•—  > 

o   o  »o 

Q    t/3 

o    o    o. 

, 

^^ 

o 

0                  O 

0 

o 

tO      M       QN 

Q 

S-fl 

u|2 

XX 

1 

5 

CN 

8 

•<}• 
t^ 

JN" 
M 

f.               l-l 

00 

0       NO 

•*      •* 

ON 
1 

! 

"  "  1 

*^J 

M 

Ml 

M 

a 

«u         p 

i    ,*o  d 

^^ 

0 

0                    0 

0 

1  £ 

•O          0 

i-'x'S^  -^ 

M    M 

o 

Y_ 

;M 

g 

8         £ 

00         "> 

!? 

o 

£-3 

O 

CS|oa 
to  ioo,n  eg 

rt   U3 

XX 

k 

0> 

* 

00 

^ 

00 
H 

•<*•               >H 

0 

N        00 
«0         ^ 

a 

0 

1  si 
SB| 

k}  ^  H 

,    . 

*  : 

0 

o 

III 

PQ 

w^'S 
8-3.8 

x  : 

1 

? 

CN 

o 

8 

00 

ON 

to 

00 

III 

°£fe 

ij 

H 

00 

< 

Clear, 
straight- 
grained. 

XX 

1 

00 

LO 

OO 

0 

o 

1 

l-~ 

•t 

O                    O 

1    1 

CO 

r>. 

NO               0> 

>0         ON 

»o 

0 

ON 

| 

§ 

1 

.         ,         • 

si 

1 

a 

: 

:    *|i 

•1       4 

Hi 

£ 

•     , 

0^,0 

Q, 

09 

w'S     • 

£           T> 

wS 

qj 

•     » 

cx^^ 

rt 

O 

*&  ^ 

•°           G 

JB.S 

jj                0) 

1 

Description  of  Lum 

|  : 

'E?S 
OfC 

1 

c 

0) 

| 

Max.  fibre  stress,  Ibs 
in.,  including  wt.  i 

Deflection,  load  first 

Deflection  at  end  of 

E  (immediate)  

E  (final)  

Modulus  of  Rupture, 
sq.  in.,  including 
of  beam  

Weight  per  cu.  ft.  a1 
ning  Ibs  

Weight  per  cu.  ft.  at 

1  Date  of  testing.  .  .  . 

I  E  (after  seasoning) 
sq.  in.  of  final  sec 

TRANSVERSE   STRENGTH  OF   TIMBER. 


697 


TABLE   OF    BEAMS   WHICH    GAVE  WAY   BY    LONGITUDINAL 

SHEARING. 


Spruce. 

Yellow  Pine. 

Oak. 

White  Pine. 

No. 

Intensity  of 
Shear, 
Lbs.  per  Sq.  In. 

No. 

Intensity  of 
Shear, 
Lbs.  per  Sq.  In. 

No. 

Intensity  of 
Shear, 
Lbs.  per  Sq.  In. 

No. 

Intensity  of 
Shear, 
Lbs.  per  Sq.  In. 

22 

202 

3° 

273 

109 

152 

129 

i55 

24 

190 

32 

242 

269 

379 

134 

"9 

31 

154 

33 

153 

233 

180 

35 

117 

5° 

177 

36 

248 

264 

46 

233 

4*9 

207 

90 

273 

429 

244 

304 

130 

437 

289 

321 

233 

45* 

151 

. 

416 

215 

452 

240 

421 

199 

423 

207 

445 

133 

45° 

,66 

' 

* 

Average,  200. 

Average,  224. 

Average,  266. 

Average,  151. 

COMPRESSION   OF  TIMBER   AT   RIGHT  ANGLES   TO   THE   GRAIN. 

On  page  698  will  be  found  a  table  giving  the  averages  of  a 
series  of  tests  of  compression  of  timber  at  right  angles  to  the 
grain. 

The  pieces  of  timber  tested  were  all  about  13"  long  in  the 
direction  of  the  grain,  the  other  two  dimensions  being  as  given 
in  the  table,  the  pressure  being  applied  in  the  direction  of  the 
longer  of  these  two  dimensions. 

In  the  cases  given  in  the  tables  a  maximum  load  was  found. 
Evidently,  however,  as  the  ratio  of  length  in  the  direction  of 
the  pressure  to  least  diameter  decreases,  we  reach  a  point 
where  no  maximum  load  is  found,  but  where  continuous 
crushing  goes  on,  the  pressure  continuously  increasing.  Thus, 
in  the  case  of  spruce  specimens  10"  X  12",  4"  X  6",  4"  X  8", 
and  6"  X  8",  while  a  maximum  load  was  sometimes  found,  at 
other  times  it  was  not. 


698 


APPLIED    MECHANICS. 


COMPRESSION    OF    TIMBER    AT    RIGHT    ANGLES    TO   THE 

GRAIN. 


5 

. 

i& 

5 

vi 

1! 

1 

u 

1 

Bfija 
0  c  £ 

"d 
o 

i   . 

U 

h 

ifi 

Approximal 
mensions. 

Number  of 

u  S>5 

eft 

etf  0  3 
I"5" 

o 

•o 
c 

Approxima 
mensions. 

Number  of 

Ibs. 

2$        10 

60 

Ibs. 

396 

r 

Ibs. 

4X  12 

60 

Ibs. 

535 

2 

21 

252 

6xr2 

16 

402 

Spruce.     ...    • 

3 
4 

23 

59 

350 
397 

Yellow  pine..  -{ 

I 

8x  12 

8x    8 

18 
5 

471 
696 

6 

26 

319 

L 

4x    6 

17 

566 

8 

31 

350 

r 

4x12 

60 

898* 

Maple  
Hemlock  

4    x  12 
4   x  12 

ai 
ao 

1808 
33° 

Oak  -j 

4x12 
6x  12 

8  X  12 

5' 
17 

21 

980* 

»35 

'913 

*  Two  different  lots. 


6°.  Framing-Joints.  —Another  matter  intimately  connected 
with  the  strength  of  timber  beams  is  the  strength  of  the  beam 
after  it  has  been  cut  in  some  of  the  various  ways  commonly 
employed  in  framing.  We  are  often  told  that  a  notch  cut  on 
top  of  a  beam,  or  at  the  middle  of  its  depth,  or  near  the  sup- 
port,  does  but  little  injury ;  but  the  tests  made,  show  the  injury 
to  be  very  large,  amounting  to  a  reduction  of  the  strength  of 
the  beam  to  one-fourth  or  one-fifth  of  its  original  strength, 
with  some  of  the  most  approved  framing. 

The  fact  is  that,  in  the  case  of  timber,  the  shearing-strength 
along  the  grain  is  small,  and  that,  in  almost  any  case  of  notch- 
ing timber,  as  in  a  notched  beam,  or  in  a  header,  there  is 
developed,  in  consequence  of  the  cutting,  a  tendency  to  tear 


TRANSVERSE   STRENGTH  OF   J^IMBER. 


699 


the  timber  across  the  grain,  and  the  resistance  of  timber  to  this 
kind  of  stress  is  very  small.  Moreover,  the  injury  due  to 
notching,  so  far  from  being  a  small  quantity,  is  very  large. 
Hence  it  follows  that  almost  any  cutting  does  a  great  deal 
of  injury ;  and  it  is  much  better  to  avoid  framing  whenever 
it  is  possible,  and  use  stirrup-irons  instead.  In  these  tests, 
only  two  of  the  most  approved  framing-joints  have  been 
tested;  viz.,  the  joint  known  as  the  " tusk-and-tenon,"  shown 
in  Fig.  243,  and  used  for  framing  the  tail-beams  of  a  floor  into 
the  headers,  and  the  "double 
tenon  and  joint  bolt,"  shown  in 
Fig.  244,  and  used  for  framing 
the  headers  into  the  trim- 
mers. 


\ 


FIG.  243. 


FIG.  244. 


The  arrangement  is  shown  in  plan  in  Fig.  245,  where  I  and 
2  are  the  trimmers,  3  is  the  header,  and  4,  5,  and  6  are  the 
tail-beams ;  the  latter  being  supported  at 
one  end  on  the  header,  and  at  the  other 
on  the  wall,  the  header  being  supported 
by  the  trimmers,  and  the  trimmers  being 
supported  on  the  walls  at  both  ends. 

It  is  sometimes  the  practice  to  hang 
the  header  in  stirrup  irons,  and  this  is  an 
improvement ;  but  it  is  very  seldom  that 
the  tail-beams  are  hung  in  stirrup  irons, 
and  these  tests  have  shown  the  weakening 
already  referred  to,  from  the  mortises  cut 
in  the  header  to  admit  the  tail-beams. 

A   spruce   floor   was    first   built   and 
tested,  the  following  being  a  partial  ac- 
FIG  «45.  count  of  the  test :  — 

No.  52. ' — Section  of  a  floor  between  the  trimmers.  Spruce : 
three  tail-beams,  2  inches  by  12  inches  each,  framed  into  a  31- 
inch  by  uf-inch  header;  header  in  turn  framed  into  sections 


^          1 

f 

f 

I 

>              4 

D 

\            .5 

0 

3 

\          6 

D 

r 

/                        2 

_/ 

700  APPLIED  MECHANICS. 

of  the  trimmers  by  double  tenon  and  joint-bolt,  cross-bridged  in 
two  places ;  tail-beams  framed  by  tusk-and-tenon  joint,  pinned, 
floored  over  and  furred  below ;  load  at  centre,  distributed  be- 
tween the  three  tail-beams  by  bridging. 

Span  =  1 6  feet ;  weight  of  joist,  flooring,  etc.,  =  331  Ibs. 

11238  Ibs.  =  breaking-load. 

Joist  on  east  side  broke  by  splitting  off  at  the  tenon,  bore 
7988  Ibs.  after.  The  load  was  then  increased.  Centre  tail-beam 
broke  by  tension  at  9988  Ibs.,  on  account  of  cross-grain  in  the 
lower  fibres.  A  split  also  started  at  the  lower  tenon  of  the 
header,  which  at  the  time  of  breaking  was  rapidly  increasing. 

Average  modulus  of  rupture  of  the  tail-beams,  including 
their  own  weight,  etc.,  =  3801  Ibs.  per  square  inch. 

Average  modulus  of  elasticity  of  tail-beams  =  1399141  Ibs. 
per  square  inch. 

It  is  to  be  noticed,  that  the  header  already  began  to  crack 
when  the  tail-beams  broke,  and  hence  that  the  floor  could  have 
borne  but  little  more,  even  if  the  load  had  been  uniformly  dis- 
tributed :  hence  that,  -in  this  case,  the  breaking-strength  of  the 
floor  would  be  determined  by  calculating  the  loads  at  the  centre 
of  the  tail-beams,  instead  of  accounting  it  as  distributed ;  in 
other  words,  the  breaking-weight  would  be  about  one-half  what 
we  should  get  by  considering  the  load  as  distributed  on  the 
tail-beams. 

YELLOW-PINE    HEADERS. 

A  number  of  tests  of  the  strength  of  yellow-pine  headers, 
and  also  of  spruce  headers,  have  been  made  in  the  Laboratory 
of  Applied  Mechanics  of  the  Massachusetts  Institute  of  Tech- 
nology, and  the  results  will  be  given  here. 

It  will  be  seen  from  these  tests  that  the  first  of  these  head- 
ers had  for  its  breaking-weight  13163  Ibs.,  and  the  second 
11631,  or  in  each  case  one-half  the  load  on  the  floor.  To 
institute  a  comparison,  we  may  observe  that,  if  a  6-inch  by 
12-inch  yellow-pine  header  6  feet  8  inches  long,  with  four  tail- 


TRANSVERSE  STRENGTH  OF  TIMBER. 


701 


beams  18  feet  long,  were  to  support  a  floor,  the  floor  surface 
would  be  96  square  feet,  giving  48  square  feet  to  be  supported 
by  the  header.  This,  if  the  floor  were  loaded  with  100  Ibs.  per 
square  foot,  would  bring  upon  the  header  4800  Ibs.,  or  about 
one-half  the  breaking-weight  of  a  header  only  5  feet  4  inches 
long  ;  whereas,  it  would  commonly  be  supposed,  that,  with  such 
a  construction  for  100  Ibs.  per  square  foot  of  floor,  we  should 
have  provided  an  unnecessarily  large  margin  of  safety. 

The  special  source  of  weakness  in  the  header  is,  of  course, 
the  joint  by  which  the  tail-beams  are  attached  to  it,  while  the 
framing  of  the  header  into  the  trimmer  causes  great  loss  of 
strength  in  the  trimmer.  Nevertheless,  even  for  the  sake  of 
the  header,  hanging  it  in  stirrup-irons  on  the  trimmer  is  better 
than  framing. 

The  fact,  also,  that  a  6-inch  by  12-inch  yellow-pine  beam  5 
feet  4  inches  long  bore  48000  Ibs.  centre  load,  equivalent  to 
96000  distributed,  without  breaking,  while  the  header  broke  at 
10916,  shows  what  an  enormous  weakening  is  caused  by  cutting 
mortises,  and  how  much  strength  would  be  gained  by  avoiding 
all  framing,  and  using  stirrup-irons  to  support  the  tail-beams  in 
all  cases  where  they  cannot  be  supported  on  top  of  the  header 
bearing  the  latter. 

TESTS   OF  YELLOW-PINE   HEADERS. 


FIG.  246. 


The  yellow-pine  headers  were  all  6  inches  wide  by  12 
inches  deep,  and,  in  every  case,  the  tail-beams  were  placed 
16  inches  apart  centre  to  centre,  so  that  the  length  of  the 


702  APPLIED    MECHANICS. 

header  between  the  trimmers,  in  any  case,  can  be  found  by 
multiplying  sixteen  inches  by  one  more  than  the  number  of 
tail-beams.  The  headers  were  either  framed  into  the  trim- 
mers by  a  double  tenon  and  joint-bolt,  or  else  they  were  hung 
from  them  by  stirrup-irons,  the  trimmers  being  supported  upon 
jack-screws.  The  headers  were  mortised  at  the  proper  places 
(sixteen  inches  on  centres)  for  twelve-inch  yellow-pine  tail- 
beams  sometimes  ten  feet  length  between  headers,  sometimes 
six,  but  more  often  three  feet  between  the  headers. 

The  load  was  applied  at  the  centre  of  the  tail-beams,  and 
divided  equally  among  them  by  iron  bars  and  knife  edges. 

The  longer  tail-beams  were  cross-bridged  by  2"  X  3" 
spruce  bridging,  but  the  shorter  ones  were  not  bridged. 

The  mortises  in  the  headers  were,  in  every  case,  those  suit- 
able for  a  tusk  and  tenon  joint,  the  tail-beams  being  in  most 
cases  three  inches  wide. 

The  table  giving  the  results  of  the  tests  of  the  yellow-pine 
headers  will  be  found  on  page  703. 

In  order  to  form  an  adequate  conception  of  the  amount  of 
the  loss  of  strength  of  one  of  these  yellow-pine  headers  carry- 
ing three  tail-beams,  assume  the  same  beam  with  no  notches, 
but  with  the  load  equally  divided  at  three  points  sixteen  inches 
on  centres,  compute  the  breaking  strength  of  such  a  beam, 
and  compare  with  it  the  strength  of  any  one  of  the  headers 
tested  which  carried  three  tail-beams.  Use  for  modulus  of 
rupture  5000  pounds  per  square  inch. 

Performing  the  calculation,  we  easily  obtain  for  the  break- 
ing strength  of  the  six-inch  by  twelve-inch  yellow-pine  beam 
without  the  notches  67500  pounds.  The  average  of  the  break- 
ing strength  of  the  four  headers  carrying  three  tail-beams  is 
13127  pounds,  which  is  only  0.194  of  67500  pounds;  hence 
the  notches  for  the  tusk  and  tenon  joints  where  the  tail-beams 
were  attached  to  the  headers  caused  a  loss  of  about  eighty  per 
cent  in  the  strength  of  the  latter. 


TRANSVERSE   STRENGTH  OF   TIMBER. 


703 


The  following  table  shows  the  results  of  the  tests  of  the 
yellow-pine  headers. 

YELLOW-PINE    HEADERS. 


Number 
of 
Test. 

Width 
(inches). 

Depth 
(inches). 

Length  be- 
tween Trim- 
mers (inches). 

Number 
of 
Tail  Beams. 

Breaking  Load 
of  each  Header. 
Pounds. 

89 

6 

12 

64 

3 

I3T63 

105 

6 

12 

64 

3 

11631 

106 

6 

12 

64 

3 

I5I3I 

107 

6 

12 

64 

3 

12581 

482 

6 

12 

80 

4 

I5I90 

483 

6 

12 

80 

4 

20190 

484 

6 

12 

96 

5 

19870 

485 

6 

12 

96 

5 

16045 

486 

6 

12 

96 

5 

16925 

487 

6 

12 

112 

6 

I86IO 

490 

6 

12 

128 

7 

13905 

492 

6 

12 

80 

4 

12795 

493 

6 

12 

112 

6 

I30IO 

500 

6 

12 

96 

5 

16370 

DETAILS    OF    THE    TESTS. 

No.  89. — The  headers  were  framed  into  the  trimmers  by 
double  tenons  and  joint-bolts. 

The  tail-beams  were  each  3  inches  by  12  inches,  and  at  first 
were  10  feet  long,  the  result  being  that,  at  a  total  load  of 
24866  pounds,  i.e.,  12433  pounds  on  each  header,  one  of  the 
tail-beams  broke  under  the  tenon  by  splitting,  while  the  head- 
ers were  left  intact.  These  tail-beams  were  then  removed 
and  new  ones  were  supplied  each  3  inches  by  12  inches  as  be- 
fore, but  only  80  inches  long,  and  the  test  was  repeated,  re- 
sulting in  the  breakage  of  one  of  the  headers  by  splitting 
through  the  middle,  following  the  line  of  mortises. 

No.  105. — The  headers  were  framed  into  the  trimmers  by 
double  tenons  and  joint-bolts.  The  tail-beams  were  each  3 
inches  by  12  inches,  and  72  inches  long. 


/04  APPLIED    MECHANICS. 

One  of  the  headers  failed  through  the  line  of  mortises. 

No.  106 — The  headers  were  supported  on  the  trimmers  by 
means  of  stirrup-irons. 

The  tail-beams  were  3  inches  by  12  inches,  and  72  inches 
long. 

At  a  total  load  of  26,662  pounds,  i.e.,  a  load  on  each  header 
°f  ^SS1  pounds,  one  of  the  tail-beams  split  below  the  line 
of  mortises.  The  total  load  was  then  increased  to  30,262 
pounds,  i.e.,  15131  pounds  on  each  header,  when  one  of  the 
stirrup-irons  broke,  but  simultaneously  with  this  the  header 
failed. 

No.  107. — The  unbroken  headers  of  Nos.  105  and  106 
were  used  for  this  test,  supported  on  the  trimmers  in  stirrup- 
irons.  At  the  maximum  load  one  of  the  headers  failed  sud- 
denly. 

Of  the  remaining  headers  the  last  three  were  supported 
on  the  trimmers  in  stirrup-irons,  the  other  seven  being  framed 
into  the  trimmers. 

No.  482. — One  of  the  headers  split,  starting  at  the  lower 
tenon  at  one  of  the  trimmers. 

No.  483. — One  of  these  headers  was  the  unbroken  one  of 
No.  482.  This  header  broke  in  the  same  way  as  in  the  case  of 
No.  482. 

No.  484. — One  of  the  headers  split. 

No.  485. — One  of  the  headers  split. 

No.  486. — One  of  the  headers  was  the  broken  one  of  No. 
483,  and  this  one  failed  by  splitting. 

No.  487. — One  of  the  headers  split. 

No.  490. — The  splitting  of  one  of  the  headers  was  followed 
almost  immediately  by  that  of  the  other. 

No.  492. — The   failure  of   one  header  was  soon  followed  by 
that  of  the  other. 

No.  493. — One  of  the  headers  failed,  splitting  very  much. 

No.  500. — One  of  the  headers  split. 


TRANSVERSE   STRENGTH  OF   TIMBER. 


70S 


TESTS  OF  SPRUCE  HEADERS. 

The  general  dimensions  of  the  floors  tested  are  shown  in 
the  figure.     In  all  these  tests  the  tail-          rp&%%  i/ >  n 

beams  are  joined  to  the  headers  by  tusk 
and  tenon  joints.  The  load  was  dis- 
tributed equally  at  the  centres  of  the 
three  tail-beams. 

The  following  table  shows  the  results 
of  these  tests. 


u 

*c* 

v 

, 

5'x.3 
(3) 

t  .L 

d 

V 

SPRUCE   HEADERS   (Figure,  page  705). 


Number 
of 
Test. 

Width 
(inches). 

Depth 
(inches). 

Length  be- 
tween Trim- 
mers (inehes). 

Number 
of 
Tail  Beams. 

Breaking  Load 
of  each  Header. 
Pounds. 

14! 

4 

12 

64 

3 

QO2O 

142 

4 

12 

64 

3 

9Q20 

143 

4 

12 

64 

3 

9415 

170  (a) 

3t 

12 

64 

3 

6917 

I7Q(£) 

3f 

12 

64 

3 

7417 

170  (c) 

31 

12 

64 

3 

9417 

SPRUCE   HEADERS  (Figure,  page  706). 

217  (A} 

4 

12 

64 

3 

1  0000 

217  (B) 

4 

12 

64 

3 

15850 

217  (O 

4 

12 

64 

3 

10450 

DETAILS    OF    THE    TESTS. 

No.  141. — Headers  were  framed  into  trimmers  by  double 
tenons  and  joint-bolts.  Tail-beams  were  made  of  spruce.  One 
of  the  headers  broke  by  tension. 

No.  142. — Headers  were  supported  on  trimmers  by  stirrup- 
irons.  Spruce  tail-beams  were  used  at  first,  but  they,  broke, 
leaving  the  headers  intact.  Then  yellow-pine  tail-beams  were 
used.  One  of  the  headers  failed  by  splitting  along  the  middle. 


706  APPLIED    MECHANICS. 

No.  143. — rThese  headers  were  the  unbroken  ones  of  Nos, 
141  and  142;  the  first  one  framed  into  the  trimmer,  the  other 
hung  from  the  trimmer  -by  stirrup-irons.  The  header  framed 
into  the  trimmer  failed  by  splitting. 

No.  170  (a). — Headers  joined  to  trimmers  by  double  tenons 
and  joint-bolts.  One  of  the  headers  split. 

No.  170  (b). — Unbroken  header  of  previous  test  used  for 
one  of  the  headers.  One  header  split. 

No.  170  (c). — The  unbroken  header  of  the  previous  test 
used  for  one  of  the  headers.  One  header  split. 

No.  217.     Dimensions  of  floor  altered   as  indicated  in  the 

nr-Ms"  15%  is"   ttVi    fiSure- 
ftg        |"|      |"|      "  .5  2I7A. — Headers  framed  into  trimmers, 

*  U  One  header  split. 

21  ?B. — Headers  supported  upon  trim- 
mers by  stirrup-irons.     One  header  split. 

2170. — Unbroken  header  of  2I7A  used 
for  one  of  the  headers.     One  header  split. 


GENERAL    REMARKS. 


The  stresses  brought  into  play  by  the  load  on  a  header 
are  not  only  bending  and  shearing,  but  also  a  tension  across 
the  grain,  and  any  method  of  figuring  the  load  a  header 
would  bear  without  taking  account  of  all  three,  and  especially 
the  latter,  would  not  furnish  correct  results. 

Moreover  the  character  of  the  results  of  tests  of  yellow- 
pine  headers  of  different  lengths,  given  on  page  703,  confirm 
this  statement,  for  the  strength  with  different  lengths  is  not 
by  any  means  universally  proportional  to  its  length,  and,  on 
the  other  hand,  the  breaking  strengths  do  not  increase  with 
the  length,  as  they  would  do  if  tension  across  the  grain  were 
the  only  stress  and  bending  did  not  take  place. 


TRANSVERSE  STRENGTH  OF  TIMBER. 


707 


BAUSCHINGER'S  TESTS. 

In  the  ninth  Heft  of  the  Mittheilungen  are  given  the  results 
of  an  experimental  study  which  Bauschinger  made  of  the 
strength  of  certain  pine  and  spruce  woods,  in  connection  with 
their  other  properties,  as  specific-gravity,  age,  time  of  felling, 
etc. ;  but  special  attention  is  given  to  the  variation  of  strength 
and  specific  gravity,  with  the  percentage  of  moisture  which  they 
contain,  i.e.,  their  condition  of  dryness.  While  he  did  a  con- 
siderable amount  of  work  upon  the  variation  of  the  tensile 
strength  with  the  percentage  of  moisture,  the  results  are  rather 
variable,  and  none  of  this  work  will  be  quoted  here  for  the 
reasons  given  on  page646, under  Tension  of  Timber.  He  him- 
self came  to  the  conclusion  that  more  satisfactory  results  could 
be  reached  by  experimenting  upon  compressive  strength. 

The  following  tables  give  summaries  of  the  tests  which  he 
made  and  reported  in  this  ninth  Heft  upon  compressive  and 
transverse  strength : 

COMPRESSIVE    TESTS. 
TEST-PIECES  ABOUT  3^x3^  INCHES  AND  6  INCHES  LONG. 


Summer  Felled. 

Winter  Felled. 

Timber. 

Place. 

Percent- 
age of 
Moisture. 

Compressive 
Strength. 
Mean  Values. 
Lbs.  per  Sq.  In. 

Percent- 
age of 
Moisture. 

Compressive 
Strength. 
Mean  Values. 
Lbs.  per  Sq.  In. 

Pine   .     . 

Lichtenhof   . 

.      9 

3997 

26 

4537 

Spruce     . 

Frankenhofen 

2O 

3449 

17 

4452 

Spruce     . 

Regenhtitte  . 

27 

3328 

20 

3997 

Spruce     . 

Schliersee     . 

2O 

2304 

T9 

3200 

708 


APPLIED  MECHANICS. 


TRANSVERSE    TESTS. 


Per- 

Modulus of 

Modulus  of 

Timber. 

Place. 

Breadth  and 
Depth. 

centage 
of  Mois- 

Elasticity. 
Lbs.  per 

Rupture. 
Lbs.  per 

Inches. 

ture. 

Sq.  In. 

Sq.  In. 

I.  Summer  Felled.     Span,  8  ft.  2.43  ins. 

Pine    .     . 

Lichtenhof  .     . 

6.70X  6.81 

24 

1422300 

6002 

•«       .     . 

i« 

7.I9X   7-17 

23 

1536084 

6685 

« 

ii 

6.66X  6.72 

19 

1536084 

6742 

« 

ii 

7.06X   7.17 

25 

1664091 

7453 

Spruce     . 

Frankenhofen  . 

6.72X  6.76 

29 

1706760 

6372 

« 

ii 

7-32X  7-25 

35 

1649868 

6344 

<  < 

<« 

7-77X  7-8o 

25 

1464969 

5703 

it 

ii 

S.ioX  8.10 

26 

1436523 

5405 

€( 

Regenhiitte  .     . 

7.52X  7-64 

39 

1578753 

6016 

« 

ii 

S.oiX  8.07 

34 

1592976 

5476 

" 

41 

7.88X  7-83 

30 

1635645 

6i73 

II 

1C 

8.36X  8.26 

32 

1720983 

6016 

tf 

Schliersee    .    . 

10.17X10.24 

25 

1016945 

4025 

n 

ii 

10.25X10.34 

24 

960052 

3840 

II 

i< 

10.47X10.47 

21 

1109394 

4281 

II 

ii 

10.73X10.67 

24 

1080948 

4281 

II.  Winter  Felled.     Span,  8  ft.  2.43  ins. 

Pine    .     . 

Lichtenhof  .     . 

6.58X  6.71 

33 

1422300 

6813 

"       .     . 

(C 

6.23X  6.30 

33 

1664091 

7609 

"       .     . 

"           .     . 

7-I9X  7-34 

31 

1308516 

5348 

ii 

II 

8.30X  8.17 

34 

1479192 

5903 

Spruce     . 

Frankenhofen  . 

7.07X  6.94 

31 

1479192 

5959 

ii 

ii 

7.IQX  7-29 

28 

1436523 

5903 

« 

<i 

6.65X  6.69 

24 

1863213 

6969 

K 

ii 

6.64X  6.67 

24 

1820544 

6813 

<« 

Regenhiitte  .     . 

6.38X  6.39 

27 

1479192 

6230 

(i 

1C 

7.02X  7-o8 

29 

1536084 

6329 

ii 

"           .     . 

6.66X  6.79 

30 

1635645 

6443 

4i 

« 

7.65X  7-66 

38 

1635645 

6358 

(i 

Schliersee    .     . 

10.98X12.66 

26 

1024056 

3641 

ii 

<i 

10.83X13-18 

25 

981387 

3755 

i« 

K 

11.19X11.23 

28 

967164 

3670 

ii 

"          .     . 

ii.nXii.io 

25 

981387 

3570 

TRANSVERSE   STRENGTH  OF   TIMBER.  709 


In  Heft  16  of  the  Mittheilungen,  Bauschinger  gives  an 
account  of  a  series  of  tests  of  the  crushing  and  transverse 
strength  of  the  more  important  coniferous  woods  from  the  dif- 
ferent districts  of  Bavaria. 

In  the  case  of  the  transverse  tests  the  percentage  of  moisture 
was  determined  by  experiment,  and  is  recorded  in  the  tables. 
Then  sections  were  cut  from  the  same  pieces  from  which  the 
beams  were  taken,  and  tested  for  crushing,  one  of  them  being 
rather  wet ;  one  had  somewhere  near  15  per  cent  of  moisture, 
which  Bauschinger  considers  to  be  about  the  average  dryness 
of  the  air,  and  one  was  somewhat  drier ;  and  in  each  case  the 
percentage  of  moisture  is  determined  and  recorded. 

The  results  of  the  crushing  tests  are  then  plotted,  and  a 
curve  drawn,  from  which  he  determines  the  crushing-strength 
with  15  per  cent  of  moisture.  A  similar  proceeding  is  adopted 
in  regard  to  the  specific-gravity. 

The  45  sections  were  each  cut  into  five  specimens  about  3 
or  4  inches  square,  one  of  them  containing  the  heart. 

These  (which  contain  the  heart)  he  omits  from  his  curves 
and  calculations,  and  plots  only  the  results  of  the  others. 

His  results  are  given  in  the  following  table,  a  perusal  of 
which  will  show  that  the  moduli  of  rupture,  and  also  the  crush- 
ing-strengths, run  somewhat  higher  than  they  do  for  woods  of 
the  same  name  in  the  tests  made  at  the  Massachusetts  Institute 
of  Technology.  This,  of  course,  maybe  due  to  the  woods  that 
Bauschinger  tested  being  stronger  than  American  woods  of  the 
same  name,  but  it  is  more  probably  due  to  the  facts  that,  i°,  the 
specimens  he  used  were  rather  smaller,  and,  2°,  they  were,  on 
the  whole,  drier  than  the  American  woods  tested  at  the  Mas- 
sachusetts Institute  of  Technology. 


APPLIED  MECHANICS. 


^ 

Mean  Compressive 

2 

Strength  of 
Pieces  not  con- 

Transverse Test. 
Span,  8  ft.  2.42  ins. 

taining  Heart, 

8 

reduced  to 

1 

j. 

Name. 

Place  of 
Growth. 

|j 

a 
15* 

b 
Same 
percent 
Moisture 

'§ 

s 

'S 

Modu- 
lus of 

Modu- 
lus  of 

w 

Moisture. 

as  in 

& 

Elas- 

Rup- 

V 

ul 

Lbs.  per 
Sq.  In. 

Trans- 
verse 

|S 

4  ' 

h 

ticity. 
Lbs.  per 

ture. 
Lbs. 

"o 
6 

f 

Test. 
Lbs.  per 

11 

f| 

S3 

V 

Sq.  In. 

Sq.In. 

55 

Sq.  In. 

OQHH 

Q£ 

p« 

J 

Larch 

St.  Zeno 

0.62 

6827 

6756 

5.78 

6.  97 

15-5 

2076560 

1038* 

2 

Larch 

0.67 

7353 

6784 

5.85 

6.99 

17.2 

1692540 

10596 

3 

Pine 

0.53 

6045 

6400 

5-74 

13.5 

1834770 

9742 

4 

Spruce 

0.43 

4978 

4551 

7'?8, 

7.61 

16.6 

1479190 

7183 

5 

Pine 

0.52 

5263 

5120 

7-73 

7.70 

15-8 

1649888 

7325 

6 

Spruce 

0.45 

5405 

5334 

7-69 

9.21 

1635650 

6756 

7 

0.48 

5547 

5831 

5.8o 

6.85 

14-3 

1592980 

7965 

8 

44 

0.51 

5974 

5689 

6.87 

6.82 

16.1 

1578750 

7751 

9 

Larch 

4* 

0.59 

6685 

4978 

6.10 

6.91 

16.0 

1706760 

9245 

10 

44 

" 

0.58 

6116 

5974 

5-85 

6.91 

J5-5 

1521860 

9743 

ii 

12 

Ztirbe 
Larch 

ii 

Karlstein 

0.41 

0.61 

3200 
7965 

3271 
7752 

a 

7.00 
7.16 

14.7 
15-8 

2097898 

5191 
9529 

13 

Spruce 

44 

0-54 

5974 

5334 

6.79 

7.78 

17.4 

1592980 

7965 

14 

Pine 

44 

0-54 

5974 

6258 

5.65 

6-93 

14.4 

1905880 

10027 

1  6 

Spruce 
Larch 

4i 

0.46 
0.68 

5405 
7965 

7325 

6.78 
7.6i 

7.80 
9-34 

16.3 
17.9 

1777880 
2039670 

8107 
8605 

17 

Spruce 

44 

0-54 

6969 

6898 

5-79 

6-95 

2026780 

9387 

18 

19 

« 

iC 

0.49 
0.52 

6045 
6187 

5476 
6045 

5-86 
5-83 

7.80 
6.87 

III 
15-6 

1578750 
1692540 

8036 

20 
21 

Pine 
Larch 

II 

0.52 
0.61 

5831 
7040 

5903 
6258 

5-77 
S-8i 

7.71 
6-93 

14.9 
17.6 

I55°3°7 
1905880 

7823 
10241 

22 

Spruce 

4* 

0.51 

6471 

6187 

5.76 

7-77 

15-8 

1564530 

8107 

23 
24 

44 

Larch 

H 

Freising 

0.39 
0.61 

4267 
7823 

4054 
7538 

7.72 
5-75 

9-31 
6-94 

15-9 

1208960 
2062340 

9814 

25 

44 

4* 

0.70 

8249 

7538 

5-84 

6.91 

16.9 

2176120 

9814 

26 

Spruce 

44 

o.59 

6969 

5831 

6.87 

7.72 

18.4 

1977000 

8249 

27 

44 

41 

0.44 

54°5 

5191 

6.96 

7.62 

15-6 

1479190 

6685 

28 
29 

Pine 

ii 

0.62 

6187 
7894 

5476 
7467 

6.88 
6.86 

7-75 
7.64 

18.4 
16.9 

1550310 
2062340 

6258 
9814 

3° 

Spruce 

44 

0.48 

5974 

5689 

6.91 

7-74 

16.7 

1607200 

7609 

32 

Pine 

(4 

0-47 
0-53 

5618 
6116 

4480 
5689 

6-93 
5-83 

7.80 
6.86 

18.6 
16.8 

1763650 
1848990 

6898 
8818 

33 

44 

44 

0.49 

4907 

4551 

5-83 

6.91 

17.0 

1223180 

5974 

34 

Spruce 

41 

o.59 

6827 

6400 

6.82 

7-74 

16.7 

1948550 

9103 

11 

\\ 

41 

0.48 
0.44 

5618 
5334 

5334 

6.96 
6.96 

7.70 
7.71 

16.4 
15-8 

1678310 
1550310 

6471 
7325 

37 

44 

14 

0.49 

6258 

5974 

7.00 

7.82 

16.4 

1806340 

7467 

38 

Larch 

44 

0.58 

7325 

7112 

5-77 

6.91 

15-5 

1742320 

9387 

39 

44 

II 

0.65 

5903 

5263 

7.00 

6.96 

18.2 

1550310 

7752 

lO 

Pine 

44 

0-49 

4764 

4267 

6.96 

7.81 

17.7 

1038280 

3485 

;t 

Spruce 

** 

0.46 

6116 

5547 

6.88 

7.80 

17.0 

1280070 

5618 

42 

44 

Jnterliezheim 

0.42 

4120 

6.85 

7.71 

20.7 

1464970 

6144 

43 
44 

White  Pine 

|« 

0-39 
°-33 

4907 
3414 

4551 
3058 

6-95 
6.99 

7.80 

18.8 
18.1 

1251620 
810731 

4"5 

45 

44 

0.32 

3200 

2631 

7.00 

7*8 

21.4 

725370 

3556 

TRANSVERSE   STRENGTH  OF   TIMBER. 


711 


AVERAGE    COMPRESSIVE    STRENGTH    OF    WHOLE    SECTION    OF    LOG. 
POUNDS  PER  SQUARE  INCH. 


Time  of 
Felling. 

Pine  from 
Lichtenhof. 

Spruce  from 
Frankenhofen. 

Spruce  from 
Regenhiitte. 

Spruce  from 
Schliersec. 

i 

2 

i 

2 

i 

2 

i 

2 

Summer 
Winter 

7183 
6343 

5244 
6784 

6416 
6756 

4807 
56l8 

6287 
6343 

5319 

5348 

4570 
4779 

3M3 
4238 

i.   Tested  5  years  after  felling.  2.  Tested  3  months  after  felling. 

OTHER  FULL-SIZE  TESTS. 

References  to  other  full-size  tests  of  timber  are: 

i°.  Tests  of  Pine  Stringers,  and  Floor  Beams,  by  Onward  Bates, 
Trans.  Am.  Soc.  C.E.,  1890. 

2°.  Tests  of  White  Pine  of  Large  Scantling,  by  Prof.  H.  T.  Bovey, 
Trans.  Canadian  Soc.  C.E.,  1893. 

3°.  The  Strength  of  Canadian  Douglas  Firs,  Red  Pine,  White 
Pine  and  Spruce  (mostly  full  size).  Trans.  Canadian  Soc.  C.E.,  1895. 

4°.  The  Proc.  Fifth  Annual  Convention  of  the  Assoc.  R.  R.  Supts. 
contains,  among  others,  the  following  references :  (#)  Tests  of 
Strength  of  State  of  Washington  Timbers,  Talbot  Hart,  and  S.  K. 
Smith  ;  (6}  Tests  of  the  Northwest  and  Pacific  Coast  Timbers,  S.  K. 
Smith  and  Thurston ;  (c)  Tests  of  California  Redwoods,  Soule ; 
(d)  Old  and  new  White  Pine  Stringers,  Finley ;  (e)  Beams  of 
Douglas  Fir,  Wing,  1895. 

5°.  U.  S.  Dept.  of  Agriculture ;  preliminary  circular  issued  by 
the  Bureau  of  Forestry,  giving  outline  of  future  work,  1903. 

6°.   U.  S.  Dept.  of  Agriculture ;  Progress  Report  on  the  Strength 
of  Structural  Timbers,  Bureau  of  Forestry,  1904. 

§  240.  Shearing  of  Timber  along  the  Grain. — The  shear- 
ing of  timber  almost  takes  place  along,  and  not  across,  the  grain; 
for  it  can  be  shown,  'that,  wherever  we  have  a  tendency  to  shear 
on  a  certain  plane,  there  is  an  equal  tendency  to  shear  on  a  plane 
at  right  angles  to  it.  Hence  if  there  is,  at  any  point  in  a  piece 
of  wood,  a  tendency  to  shear  across  the  grain,  there  must  neces- 
sarily accompany  it  an  equal  tendency  to  shear  it  along  the  grain; 
and  wherever  (as  is  almost  always  the  case)  the  resistance  to  the 
latter  is  less  than  the  resistance  to  the  former,  the  timber  will  give 
way  in  this  manner,  instead  of  across  the  grain.  As  to  the  shear- 
ing-strength per  square  inch,  some  values  have  been  given  in  Ran- 


7I2 


APPLIED   MECHANICS. 


kine's  table ;   and  the  following  table  contains  results  obtained  at 
the  Watertown  Arsenal,  and  recorded  in  Tests  of  Metals  for  1881. 


Kind  of  Wood. 

Arsenal 
No. 

Shearing- 
Strength 
per  Square 
Inch. 

Kind  of  Wood. 

Arsenal 
No. 

Shearing- 
Strength 
per  Square 
Inch. 

Ash    

62O 

600 

Oak  (white)     .    . 

631 

7^2 

621 

592 

Pine  (white)    .    . 

o 

752 

/  j" 
324 

622 

458 

753 

267 

623 

700 

754 

352 

Birch  (yellow)      . 

623 

563 

755 

366 

633 

8I5 

Pine  (yellow)  .    . 

607 

399 

634 

672 

608 

3i7 

635 

612 

614 

409 

Maple  (white)  .    . 

636 

647 

615 

4i5 

637 

537 

616 

409 

638 

367 

617 

364 

639 

43i 

618 

286 

Oak  (red)    .    .    . 

624 

775 

619 

330 

625 

743 

Spruce   .... 

748 

253 

626 

999 

749 

374 

627 

726 

75° 

347 

Oak  (white)    .    . 

628 

966 

75i 

316 

629 

803 

Whitewood     .    . 

609 

406 

630 

846 

610 

382 

§  241.  General  Remarks.  —  A  perusal  of  the  tests  on 
columns  and  on  beams  will  show  that  one  of  the  principal 
sources  of  weakness  in  timber  is  the  presence  of  knots,  and  it 
will  be  noticed  that  the  position  of  the  fracture  is  in  most 
cases  determined  by  the  knots. 

Sap-wood,  season  cracks,  and  decay  are  doubtless  other 
sources  of  weakness.  The  tests,  however,  do  not  present  such 
striking  evidence  of  the  deleterious  effects  of  the  first  two  as 
is  the  case  with  knots.  In  general,  it  may  be  said,  however, 
that  timber  used  in  construction  should  be  free,  or  nearly  free, 
from  sap-wood ;  as  an  excessive  amount  of  sap-wood  renders  it 
weak. 


GENERAL    REMARKS.  713 

It  will  often  be  found  to  be  a  common  opinion  among  lum- 
ber-dealers, that  a  piece  of  timber  which  contains  the  heart  is 
not  as  good  as  one  which  is  cut  from  the  wood  on  one  side  of 
the  heart.  This  is  very  often  true  ;  as  the  timber  which  is  sold 
in  the  market  is  very  liable  to  have  cracks  at  the  heart,  and 
also,  if  the  tree  has  passed  maturity,  the  heart  is  the  place 
where  decay  is  likely  to  begin.  Nevertheless,  the  tests  of 
beams  would  not,  it  seems  to  the  author,  bear  out  the  conclu- 
sion that  such  pieces  as  contain  the  heart  are  always  weaker 
than  those  that  do  not. 

Another  matter  that  claims  serious  consideration  is  the 
effect  of  seasoning  upon  the  strength  of  timber.  This  question 
can  only  be  decided  by  tests  on  full-size  pieces,  as  the  small 
pieces  season  much  more  rapidly  and  uniformly  than  full-size 
pieces. 

In  this  regard,  the  observation  should  be  made,  that  prac- 
tically our  buildings  and  other  constructions  are  built  with 
green  lumber;  i.e.,  lumber  which  has  been  cut  from  three 
months  to  a  year.  Unless  it  can  be  shown  that  the  seasoning 
which  the  lumber  receives  while  in  use  imparts  to  it  a  greater 
strength,  it  will  only  be  proper  to  consider  its  strength  the  same 
as  that  of  green  lumber.  Not  very  much  evidence  has  thus  far 
been  obtained  upon  this  point ;  but,  such  as  it  is,  it  will  be 
noted  here. 

i°.  We  have,  on  p.  653,  the  results  of  the  tests  of  a  lot  of 
old  mill  columns  ;  and,  while  some  of  them  did  exhibit  a  greater 
strength  than  green  ones,  a  perusal  of  this  set  of  tests  will 
convince  the  reader  that  it  would  not  be  safe  to  rely  upon  any 
greater  strength  in  these  columns  than  in  green  ones.  More- 
over, these  columns  had  been  in  a  building  heated  by  steam  for 
a  number  of  years,  and  during  the  seasoning  process  they  had 
been  subjected  to  the  load  they  had  to  support.  The  writer 
has  also  observed  some  evidence  of  the  same  kind  in  con- 
nection with  one  of  his  time  tests. 


7 14  APPLIED    MECHANICS. 


2°.  In  the  case  of  beams,  we  have,  in  Nos.  60  and  66, 
examples  of  beams  which  had  been  seasoning,  unloaded,  in  a 
building  heated  by  s-team  ;  and  in  these  cases  there  was  a  great 
gain  in  strength.  Some  yellow-pine  beams  exhibited  a  similar 
action.  On  the  other  hand,  beams  Nos.  18  and  19  had  been 
seasoning  on  the  wharf,  in  the  open  air,  for  about  one  year ; 
and  while  some  yellow-pine  beams  which  had  seasoned  without 
load,  in  the  building,  showed  great  strength,  in  other  cases  the 
increase  was  not  so  marked. 

In  view  of  the  fact  that  the  above  is  practically  all  the  evi- 
dence we  have  in  the  matter,  it  would  seem  to  the  writer, 
unless  future  experiments  shall  prove  the  contrary  to  be  true, 
that  we  cannot  rely,  in  our  constructions,  upon  having  any 
greater  strength  than  that  of  the  green  lumber,  and  that  the 
figures  to  be  used  should  be  those  obtained  by  testing  green 
lumber. 

§  242.  Building-Stones. — The  three  most  important  factors 
about  a  building-stone  besides  its  beauty,  are  its  durability,  its 
strength,  and  the  ease  with  which  it  can  be  quarried  and 
worked.  In  order  to  be  durable  it  must  be  able  to  withstand 
the  deleterious  influences  of  rain,  wind,  frost,  fire,  and  of  the 
acids  that  are  found  in  the  air  especially  in  large  -cities,  where 
the  most  common  are  carbonic  acid  and  sulphur  acids. 

The  durability  of  a  stone  is  probably  its  most  important 
feature,  and  is,  perhaps,  the  most  difficult  to  test  thoroughly. 
As  a  rule,  the  greater  its  hardness  and  the  less  its  absorptive 
power  for  moisture,  the  more  durable  will  it  be. 

Tests  of  hardness  are  easily  and  frequently  made.  Tests  of 
absorptive  power  for  moisture  are  very  often  made.  While 
the  methods  pursued  by  different  people  differ,  they  consist 
essentially  of  weighing  the  specimen  dry,  and  then  soaking  it 
in  either  hot  or  cold  water  until  it  has  absorbed  all  that  it  will, 
and  weighing  it  again. 

There  are  a  number  of  methods  pursued  in  order  to  de- 


B  U7L  DING-S  TONES.  7 1  5 


termine  its  power  of  withstanding  the  action  of  frost,  and  the 
res-ults  differ,  of  course,  according  to  the  method  pursued. 
Bauschinger's  method  consists  in — 

i°.  Determining  the  compressive  strength  of  the  stone  in 
a  dry  and  in  a  saturated  condition,  and  comparing  the  two. 

2°.  Determining  the  compressive  strength  after  twenty- 
five  freezings  and  thawings. 

3°.  Determining  the  loss  of  weight  after  these  twenty-five 
freezings. 

4°.  Examining  the  specimen  with  a  microscope  for  cracks 
after  the  twenty-five  freezings. 

Stones  will  not  withstand  the  heat  of  a  large  conflagration, 
brick  being  better  than  any  building-stone. 

As  to  how  well  a  stone  will  stand  the  gases  in  the  air  of  a 
large  city,  a  great  many  tests  have  been  proposed  and  used,  but 
none  of  them  are  entirely  satisfactory.  Of  course  we  can  get 
indications  from  a  study  of  the  chemical  composition,  or  better 
from  a  microscopical  examination,  which  shows  also  the 
arrangement  of  the  different  components,  this  being,  of  course, 
the  part  of  the  geologist  or  mineralogist. 

After  the  question  of  durability,  the  strength  comes  in  as 
the  factor  of  next  importance,  and  although  the  loads  usually 
put  upon  stones  in  construction  are  very  much  smaller  than 
the  breaking-strength  of  the  stone  as  shown  by  small  speci- 
mens tested  in  the  testing-machine,  nevertheless  the  mortar 
or  cement  joints,  the  bonding,  and  the  necessary  unevenness 
render  the  real  factor  of  safety  very  much  less  than  would  be 
at  first  imagined. 

Building-stones  are  more  often  called  upon  to  bear  a 
compressive  load  than  any  other,  though  they  are  sometimes 
called  upon  to  bear  a  transverse  load,  as  in  the  case  of  window- 
lintels. 

The  results  are  very  variable,  partly  because  the  stone 
vanes  very  much  in  quality,  and  partly  because  it  is  only  of 


APPLIED   MECHANICS. 


late  years  that  it  has  been  recognized  that  in  order  to  obtain 
correct  results  in  compression  tests  the  pressure  must  be  evenly 
distributed  over  the  surfaces  of  the  specimen  pressed  upon, 
and  that  in  order  to  accomplish  this  even  distribution  it  is 
necessary  that  the  faces  which  come  in  contact  with  the  plat- 
forms of  the  testing-machine  shall  be  accurate  planes,  and  that 
unless  the  compression  platforms  are  adjustable,  the  two  faces 
pressed  upon  shall  be  parallel  —  provided,  of  course,  the  plat- 
forms are  parallel,  as  they  should  be.  Formerly  it  was  thought 
that  the  desired  result  could  be  obtained  by  interposing  be- 
tween the  surface  of  the  specimen  and  the  platform  some 
soft  substance,  as  a  cushion  of  wood  or  of  lead,  whereas  it  is  a 
fact  that  any  such  cushions  only  render  the  results  smaller 
and  more  variable  than  the  real  crushing-strength  of  the  speci- 
mens. Hence  it  is  that  a  great  many  of  the  tests  that  have 
been  made  are  of  no  value,  because  this  matter  was  not 
attended  to.  In  a  rough  way  we  may  divide  the  most  com- 
mon building-stones  as  follows  :  i°.  Granites  and  allied  stones  ; 
2°.  Limestones,  including  marbles  ;  3°.  Sandstones  ;  4°.  Slates. 

The  most  systematic  set  of  tests  was  made  by  Bauschingerr 
and  is  reported  in  the  Mittheilungen,  Hefte  I,  4,  5,  6,  10,  II, 
1  8,  and  19. 

Besides  this  we  may  note  the  following  references  : 

i°.  Report  on  Compressive  Strength,  etc.,  of  the  Building-Stones  in 

the  United  States,  1876.     Gillmore. 
2°.  Compressive  Resistance   of    Freestone,  Brick   Piers,  Hydraulic 

Cements,  Mortars,  and  Concretes,  1888.     Gillmore. 
3°.   Masonry  Construction,  1889.     Baker. 
4°.  Testing  Materials  of  Construction,  1888.     Unwin. 
5°.  History  of  the  St.  Louis  Bridge,  1881.     Woodward. 
6°.  Exec.  Doc.  12,  47th  Congress,  ist  session.     Senate. 
7°.  Exec.  Doc.  35,  49th  Congress,  ist  session.     Senate. 

Some  tables  of  results  will  now  be  given. 


B  UILD 1NG-S  TONES. 


717 


The  following  is  taken  from  Woodward's  "  History  of  the 
St.  Louis  Bridge :" 


Material. 

Length, 
in 
inches. 

Diameter  of 
Cross-section, 
in  inches. 

Modulus  of 
Elasticity,  • 
pounds  per 
square  inch. 

Breaking- 
Weight,  per 
square  inch. 

Grafton  Magnesian  Limestone. 

41                                           it                                          « 
(1                                          «                                          II 
«                                          «                                           <l 

"               "     (5  specimens) 

6.46 
5.37 
5.96 
5-99 
•3  oo 

1.14 
1.  06 
1.  06 
1.07 
q  V  3 

10500000 
8400000 
8500000 
6OOOOOO 

7200 
8500 
2000 
6000 
av  1^400 

«               « 
Portland  Granite  

8.00 
13.00 

=;  88 

2.38 
I-I3 
2.  ^6 

I2OOOOOO 
5000000 
55OOOOO 

IOIOO 

10800 
16000 

e    08 

2    76 

64OOOOO 

18500 

«                it 

e.    Q7 

2.38 

5OOOOOO 

17000 

6  oo 

2.  ^O 

I35OOOOO 

16400 

Portland  Granite  

o    OO 

•3     V     3 

I37OO 

Missouri  Red  Granite  

^.OO 

T  X  3 

I27OO 

«                     « 

•j.QO 

a  v  "? 

I3OOO 

«                     it 

•5     OO 

•3   V    q 

I27OO 

>  i                     « 

300 

q   y   o 

13600 

Brown  Ochre  Marble 

300 

•3  y  i 

I5OOO 

Sandstone,  St   Genevieve   Mo. 

300 

•3     V     ^ 

caqo 

it                        (i 

4  88 

4  88  X  4  88 

CCQO 

«                       (i 

•i  06 

3  06  X  3  06 

ajxx) 

In  Heft  4  of  the  Mittheilungen,  Bauschinger  gives  a  long 
table  of  results  of  testing  granites,  limestones,  and  sandstones, 
from  which  the  following  examples  are  selected : 

In  Heft  5  of  the  Mittheilungen  is  to  be  found  a  study  of 
the  modulus  of  elasticity  of  building-stones.  Bauschinger 
found  that  in  this  case  the  departure  from  Hooke's  law  is 
greater  with  small  than  with  large  loads.  Heft  6  contains  an 
experimental  study  of  the  laws  of  compression.  Heft  10  con- 
tains an  experimental  investigation  of  the  principal  Bavarian 
building-stones.  Hefte  u  and  18  contain  a  study  of  the  com- 
pressive  strength  and  wearing  qualities  of  paving-stones.  Heft 
19  contains  a  study  of  the  power  of  stones  to  resist  frost.  For 
all  these  the  student  is  referred  to  the  Mittheilungen. 


7i8 


APPLIED  MECHANICS. 


Kind  of  Stone. 

Place. 

Tests  by  Compres- 
sion. 

Transverse  Tests, 

Crushing 
Strength, 
pounds 
per  sq.  in. 

Direction 
of  the 
Pressure 
with  re- 
spect to 
the  bed. 

Modulus 
of  Rup- 
ture, 
pounds 
per  sq.  in. 

Direction 
of  Break- 
ing Sec- 
tion with 
respect  to 
the  bed. 

Granites. 
Yellow,  very  soft,  exceptional  quality 

Gray    coarse-grained    

Selb  in  Ober- 
franken. 

Waldstein 
St.  Gotthard 

Cham 
Ebendaher 

Schlanders  in 
Tyrol 
Wurzburg 
Kronach 

Kelheim 

7750 
11300 
11720 

14790 
11740 
12660 
15650 
13230 

22190 
19200 

12800 
6260 
22760 
11390 

(     6900 
\     8390 
3560 
{10600 
20340 
17920 

j    18490 
1    16780 
j    12520 
j    21240 

4836 

19270 
20550 

11950 
8100 
j     8620 
1     9390 
654C 

2420 

2490 
4340 

5480 
2680 

•i      4f° 

1      3630 

1 
1 
II 

i 

\ 
\ 

i 

1 

i 

i 

1 
II 

1 

\ 

1 

\  i 

II 

1 

1 
i 

1 

I 

1 
II 

av.i94o 
1309 
2773 

1991 
980 

1252 
939 

j-  0560 

oblique 
across 

Very  light-colored,  tolerably  coarse- 

Very  hard,  coarse-grained    .... 
Very  hard   striped         .... 

White,    very     hard,    tolerably  fine- 
grained, only  good  for  paving  . 
Syenite,  black,  with  a  good  deal   of 

Limestones. 
White  marble             

Muschelkalk          

u 

Yellowish  white,  soft  limestone  of  the 

Poorest  quality  
Granitic  marble         ....... 

Rosenheim 

Poppenheim 
Herzbruch 

Kronach 
Unterfranken 

Carlsruhe 
Wurtemburg 

Ansbach 

Nuremberg 
Regensberg 

Kelheim 

Dolomite. 
From  the  white  Frankenjura.    .    .    . 

Sandstone. 
BunterJJsandstone,  gray,  with  yellow 
and  brown  streaks       ... 

Bunter  sandstone,  red,   very  rich  in 
quartz,  fine-grained     

Do.               do.               do. 
Bunter    sandstone,  dark    red,    fine- 

Do,               do.               do. 
Keuper  sandstone,  red,  fine-grained. 

Do.              do.              do. 
Keuper  sandstone,  white,  with  red- 
dish bed  stripes,  coarse-grained. 
Keuper    saudstone,  white,  tolerably 
fine-grained  

Keuper  sandstone,  brown      .     .    .     . 
Green  sandstone,  yellow,  with  brown 
layers  fine-grained      

Green  sandstone,  dirty  green,  tolera- 
bly fine-grained  

Green     sandstone,     greenish,    fine- 
grained    

B  UILDING-S  TONES. 


719 


The  following  table  is  taken  from  the  Trans.  Am.  Soc.  Civ. 
Eng.  for  Oct.  1886,  where  it  is  quoted  from  "  Mechanical  tests 
of  building  materials,  made  at  the  Watertown  Arsenal,  by  the 
United  States  Ordnance  Dept.,  at  the  request  of  the  Commis- 
sioners for  the  erection  of  the  Philadelphia  Public  Buildings :" 


Kind  of 
Stone. 

Locality. 

Color. 

Direction  of 
Pressure. 

Total 
Load 
applied, 
Ibs. 

Crushing 
Strength 
per  sq.  in. 
applied, 
Ibs. 

Sectional 
Area, 
sq'  in. 

Remarks. 

Lee,  Mass  

Blue. 

End. 

715000 

20504 

34-8? 

Burst  in  fragments. 

"       "        .... 

White. 

Bed. 

800000 

22370 

35-i6 

Slight  flaking. 

"       "        .... 

W.  &B. 

End. 

800000 

22860 

34-99 

No  apparent  injury. 

it       it 

White. 

" 

800000 

22820 

35-05 

it         ii              ti 

"       "        .... 

Blue. 

Bed. 

800000 

22900 

34-93 

Flaked,  one  edge. 

3 

d       it 

W.  &B. 

" 

767000 

21700 

35-34 

Crushed  suddenly. 

*' 

Montgomery  Co.,  Pa. 

Blue. 

44 

466300 

11470 

40  64 

Failed  suddenly. 

2 

u                    ii 

44 

End. 

400000 

10420 

38.40 

Ultimate  strength. 

tt                    u 

ti 

Bed. 

543000 

13700 

39-63 

it              it 

ti                    ii 

ii 

End. 

398000 

IOI20 

39-33 

tl                                1C 

ti                    u 

ti 

" 

347500 

959° 

36.24 

tl                  It 

, 

il                                           11 

11 

Bed. 

434000 

10940 

39-67 

It                  11 

Conshohocken  Pa. 

End. 

oe   QC 

Ultimate  strength. 

f 

494000 

14090 

35  tU*> 

.  i 

ii             u 

Bed. 

cfifvwi 

-r 

3A     67 

<               ti 

c 
o 

Indiana    ..... 

End. 

777OOO 

IUJ4O 
853O 

J4  *  uj 
44.22 

t               ii 

3//*-w-»v 
32O5OO 

Vjj^ 

7  TQO 

44.56 

i               ti 

1 

it 

Bed. 

32IOOO 

*y 

7776 

41-38 

t               it 

H* 

ii 

ii 

A  •jfi'JOO 

n/^ 

41    28 

t               it 

.  I 

43°3UU 

41  .  -io 

_« 

Vermont       .    .          J 

Dove-     } 

Bed. 

COTO/V-J 

TQ.4OO 

on  6c 

Ultimate  strength. 

S  i 

5  ji^uu 

4  J4«_>U 

oV  •  *o 

rt   j 

"             .    .         J 

colored  ' 

End. 

•370800 

0870 

08  48 

it              ti 

j/yomj 

yo/Aj 

o°  "4° 

Kind  of  | 
Stone. 

Locality. 

Color. 

Direction  of 
Pressure. 

Total 
Load 
applied, 
Ibs. 

Crushing 
Strength 
per  sq.  in. 
applied, 
ibs. 

Sectional 
Area, 
Sq.  In. 

Remarks. 

Hummelstown,  Pa.  . 
<i              tt 



Bed. 
End 

528700 

12810 

41.28 
41  92 

Ultimate  strength. 
<i             it 

Buff 

Bed 

ii             ii 

(i 

End 

*86o 

ii             11 

«J 

M 

«i 

Bed 

it             ii 

p 

it 

End 

40.06 

Bearings  imperfect 

S  • 

•o 

Blue. 

ii 

7680 

7Q    68 

Ultimate  strength 

Q 

rt 

it 

Bed 

41                             it 

W 

ti 

End 

301800 

II                             11 

,4 

Bed 

11                             il 

it 

End 

16280 

41  28 

i 

,, 

Bed 

72100 

17420 

720  APPLIED   MECHANICS. 

§  243.  Hydraulic  Cements  and  Brick  Piers. — When  a 
pure  or  nearly  pure  limestone  is  calcined,  so  as  to  drive  off  the 
carbonic  acid,  we  have  an  oxide  of  lime,  commonly  called  quick- 
lime, which,  on  the  addition  of  water,  slakes,  with  the  develop- 
ment of  considerable  heat;  and  the  result  is  a  fine  powder,  which 
by  the  addition  of  more  water  is  reduced  to  a  paste  which  slowly 
hardens  upon  exposure  to  the  air.  It  is  this  paste,  mixed  with 
sand,  which  forms  the  mortar  used  in  cheap  buildings.  It  is  very 
weak,  and  hardens  very  slowly,  even  in  the  air. 

On  the  other  hand,  a  hydraulic  lime  or  a  hydraulic  cement 
contains  impurities,  of  which  silica  forms  the  principal  portion, 
though  we  usually  find  also  alumina,  protoxide  of  iron,  and 
magnesia;  and  these  impurities  are  in  so  large  a  proportion  that 
the  slaking  entirely  or  nearly  disappears,  and  the  addition  of 
water  after  calcination  causes  the  formation  of  hydrated  silicates, 
etc.,  which  harden  under  water. 

While  we  cannot  draw  a  sharp  line  of  demarcation  between 
hydraulic  lime  and  hydraulic  cement,  nevertheless  the  essential 
difference  is  that  the  first  contains  pure  lime  in  sufficient  pro- 
portions to  slake,  but  at  the  same  time  contains  enough  clay, 
silica,  etc.,  to  enable  it  to  set  under  water;  whereas  hydraulic 
cement  contains  less  pure  lime,  and  hardly  slakes  at  all,  but 
sets  more  rapidly  than  hydraulic  lime. 

Hydraulic  cements  are  known  as,  i°,  Portland  cement,  and, 
2°,  Natural  cement.  The  latter  is  commonly  called  Rosendale 
in  America,  and  Roman  in  Europe.  It  acquires  its  strength 
more  slowly,  is  weaker  and  cheaper  than  Portland,  and  it  usually 
sets  more  quickly. 

Portland  cement  is  manufactured  extensively  in  France,  Ger- 
many, and  England,  and  in  the  United  States.  It  is  made  by 
mixing  either  dry  or  in  paste,  and  then  calcining,  to  the  point  of 
incipient  vitrefaction,  such  mixtures  of  rocks  as  will  give  the 
proper  chemical  composition.  The  paste,  when  ready  for  the 
furnace,  should  contain  from  76  to  81  per  cent  of  carbonate  of 
lime,  and  from  19  to  24  per  cent  of  clay. 


HYDRAULIC  CEMENTS  AND    BRICK  PIERS.  7 21 

To  give  precise  definitions  of  what  constitutes  Portland 
cement,  what  Natural  cement,  what  Hydraulic  lime,  etc.,  is  not 
an  easy  matter.  An  attempt  to  do  so  was  made  by  the  Inter- 
national Association  for  Testing  Materials,  but  their  definitions 
are  not  universally  accepted,  and  will  not  be  given  here. 

For  definitions  of  Portland  cement,  and  of  Natural  cement, 
which  are  by  no  means  perfect,  but  which  will  answer  in  a  general 
way,  the  reader  is  referred  to  those  adopted  by  the  Am.  Soc.  for  Test- 
ing Materials  on  pp.  730  and  733.  The  manufacture  of  Natural 
cement  dates  from  a  very  early  period,  and  depends  upon  finding 
rocks  of  suitable  composition;  that  of  Portland  cement  dates 
from  the  early  part  of  the  nineteenth  century,  when  Jos.  Aspdin, 
of  Leeds,  made  a  slow-setting  cement,  by  calcining  a  mixture  of 
carbonate  of  lime,  and  clay,  in  suitable  proportions. 

TESTS  OF  THE  STRENGTH  OF  CEMENTS. 

While  a  good  many  tests  have  been  made  on  the  compres- 
sive  strength  of  cement,  and  a  few  also  on  transverse  or  shear- 
ing strength,  nevertheless  the  test  most  commonly  used  in 
order  to  determine  its  quality  is  the  test  of  its  tensile  strength. 
The  specimen  used  for  this  purpose  is  called  a  briquette,  and 
the  cut  shows  one  of  its  common  forms,  the  smallest 
section  being  generally  one  square  inch.  The  real 
reason  for  using  the  tensile  instead  of  the  compres- 
sive  strength  for  a  test  is,  in  the  opinion  of  the 
author,  that  inasmuch  as  the  tensile  strength  of 
cement  is  very  much  less  than  its  compressive 
strength,  it  follows  that  the  machines  for  testing  the 
tensile  strength  are  cheaper,  and  the  work  of  testing  tensile 
strength  is  less  than  is  the  case  in  testing  its  compressive 
strength,  although  there  are  some  wno  give  other  reasons 
which  have  some  appearance  of  plausibility. 

In  order  to  discuss  this  matter  intelligently,  however,  we 
should  bear  in  mind  — 

Either    the    tensile   or   the   compressive  test   is  compara- 


722  APPLIED    MECHANICS. 


live  and  merely  helps  to  determine  the  quality,  and  also  to 
compare  different  lots  of  cement,  but  neither  of  them  furnish 
us  any  figures  which  would  be  suitable  to  use  in  computing 
the  allowable  load  on  any  structure  which  depended  upon 
cement  for  its  strength. 

We  might,  therefore,  conclude  that,  as  far  as  the  objects 
that  can  be  attained  by  testing  cement  are  concerned,  either 
test  would  answer  the  purpose  equally  well;  but  inasmuch 
as  it  is  also  a  fact  that,  with  the  best  appliances  thus  far 
provided  for  the  purposes,  it  is  possible  to  obtain  greater 
accuracy  in  the  compressive  than  in  the  tensile  test,  therefore 
it  seems  to  the  author  that  the  compressive  and  not  the  tensile 
is  the  test  that  should  be  used  in  making  cement  tests.  Never- 
theless, inasmuch  as  the  tensile  strength  is  most  used,  a  brief 
account  will  be  given  here,  showing  what  has  been  done,  what 
we  can  reasonably  expect  from  good  cements,  and  a  few  pre- 
cautions will  be  mentioned,  which  it  is  necessary  to  use  in 
making  the  tests,  in  order  to  insure  correct  results. 

The  literature  of  cement  testing  is  very  extensive,  but  only 
the  following  will  be  given  here 

i°.  Q.  A.  Gillmore:  Practical  Treatise  on  Limes,  Hydraulic  Cements, 
and  Mortars. 

2°.  John  Grant:  Articles  in  the  Proceedings  of  the  British  Institu- 
tion of  Civil  Engineers,  vols.  xxv.,  xxxii.,  and  xli. 

3°,  Charles  Colson:  Experiments  on  the  Portland  Cement  used  in 
the  Portsmouth  Dockyard  Extension.  Proc.  Brit.  Inst.  Civ. 
Engrs.,  vol.  xli. 

4°.  Isaac  John  Mann:  The  Testing  of  Portland  Cement.  Proc.  Brit. 
Inst,  Civ.  Engrs.,  vol.  xlvii. 

5°.  Wm.  V.  Maclay:  Notes  and  Experiments  on  the  Use  and  Testing 
of  Portland  Cement.  Trans.  Am.  Soc.  Civ.  Engrs.,  Dec.  1877. 

6°  Eliot  C.  Clarke:  Record  of  Tests  of  Cement  made  for  the  Bos- 
ton Main  Drainage  Works,  1878-1884.  Trans.  Am.  Soc.  Civ. 
Engrs.,  April,  1885. 


HYDRAULIC  CEMENTS  AND    BRICK  PIERS. 

7°.  Q.  A.  Gillmore:  Notes  on  the  Compressive  Resistance  of  Free- 
stone, Brick  Piers,  Hydraulic  Cements,  Mortars,  and  Con- 
cretes. 

8°.  J.  Sondericker:  How  to  Test  the  Strength  of  Cements.  Am. 
Soc.  Mech.  Engrs.  for  1888. 

9°.  Bauschinger:  Mittheilungen  aus  dem  Mechanisch-Technischen 
Laboratorium,  Hefte  i.,  vii.,  and  viii. 

10°.  Exec.  Doc.  12,  47th  Congress,  ist  session,  House:  Compres- 
sive Tests  of  Seven  Cubes  of  Concrete. 

11°.  Exec.  Doc.  5,  48th  Congress,  ist  session,  Senate:  Shearing 
Test  of  One  Concrete  Cube. 

12°.  Exec.  Doc.  35,  49th  Congress,  ist  session,  Senate:  Tests  of 
Neat  Cement  and  Cement  Mortars. 

13°.  Preliminary  Report  of  the  Committee  on  a  Uniform  System  for 
Tests  of  Cement.  Trans.  Am.  Soc.  Civ.  Engrs.,  January,  1884. 

14°.  Final  Report  of  the  Committee  on  a  Uniform  System  for  Tests 
of  Cement.  Trans.  Am.  Soc.  Civ.  Engrs.,  January,  1885. 

15°.  Behavior  of  Cement  Mortars  under  various  Contingencies  of 
Use.  F.  Collingwood:  Trans.  Am.  Soc.  Civ.  Engrs.,  Nov. 
1885. 

1 6°.  Report  of  Progress  by  the  Committee  on  the  Compressive 
Strength  of  Cements,  etc.  Trans.  Am.  Soc.  Civ.  Engrs.,  July, 
1886. 

17°.  Another  Report  of  the  Committee.  Trans.  Am.  Soc.  Civ. 
Engrs.,  June,  1888, 

1 8°.  E.  F.  Miller  :  Testing  Cement.     The  Bricklayer. 

19°.  Candlot,  E.  :  Ciments  et  Chaux  hydrauliques."  Fabrication, 
Proprietes,  Emploi.  Paris. 

20°.  Feret,  R.  :  Note  sur  Diverses  Experiences  concernant  les 
Ciments.  Annales  des  Ponts  et  Chaussees,  1890,  ir  semestre, 
page  313. 

21°.  Alexandre,  Paul :  Recherches  experimentales  sur  les  Mortiers 
hydrauliques.  Annales  des  Ponts  et  Chaussees,  1890,  2e  se- 
mestre  p.  227. 

22°.  Feret,  R.  :  Sur  la  compacite  de  Mortier  hydraulique.  Annales 
des  Ponts  et  Chaussees,  1892,  2e  semestre,  page  5. 


APPLIED    MECHANICS. 


23°.  D.  B.  Butler:  Portland  Cement. 

24°.  Baumaterialienkunde :  This  is  the  official  organ  of  the  Inter- 
national Assoc.  for  Testing  Materials,  and  contains  many 
papers,  and  discussions  on  cement. 

25°.  Commission  des  methods  d'essai  des  materiaux  de  construc- 
tion. Tome  i,  Section  B. — Essais  des  materiaux  d'aggre- 
gation  des  mafonneries.  Rapport  General  pre*sente  par 
Paul  Alexandre. 

26°.  Tests  of  Metals  made  at  Watertown  Arsenal. 

27°.  Mit  .  der  Materialpriifungsanstalt  in  Zurich. 

28°.  Mitt,  aus  dem  Mech.  Tech.  Lab.  in  Berlin. 

29°.  Mitt,  aus  dem  Mech.  Tech.  Lab.  in  Miinchen. 

30°.  Report  of  Board  of  Engineer  officers  on  testing  hydraulic 
cements,  1902. 

31°.  Many  articles  in  the  Trans.  Am.  Soc.  Civil  Engineers. 

32°.  Many  papers  read  before  the  Association  of  American  Portland 
Cement  Manufacturers. 

Some  quotations  will  be  given  from  Candlot's  treatise, 
including  a  portion  of  the  specifications  of  the  French  Maritime 
Service. 

Candlot  says : 

"  The  properties  of  Portland  cement,  which  have  given  complete 
satisfaction  for  many  years,  being  known,  well  defined,  and  absolutely 
constant,  it  ought  to  be  sufficient,  in  order  to  determine  the  value  of  a 
cement,  to  see  whether  it  presents,  to  the  same  degree,  the  qualities 
which  characterize  this  list  of  hydraulic  products." 

"  The  tests  generally  made  on  cements  have  to  do  with  their  chemical 
composition,  their  density,  their  fineness  of  grinding,  their  time  of  set- 
ting, their  tensile  and  compressive  strength,  and  their  invariability  of 
volume." 

The  reason  for  each  of  these  tests  is  so  plain  that  no  comment  will 
be  made,  except  to  say  that  a  cement  that  swells  is  liable  to  disintegrate 
after  setting  in  consequence  of  free  lime. 

In  France  the  greater  part  of  the  large  manufactories  are  to  be  found 
in  the  region  around  Boulogne-sur-Mer ;  and  the  maritime  service  of 
the  Department  of  "  Ponts  et  Chauss6es,"  which  has  a  cement  laboratory 
at  Boulogne,  has  established  certain  specifications  to  which  all  cement 
used  in  their  work  must  conform. 


HYDRAULIC   CEMENTS  AND    BRICK  PIERS. 

A  portion  of  these  specifications  as  given  by  Candlot  will  now  be 
quoted  in  the  following  three  pages  : 

ART.  i.  The  Portland  cement  furnished  shall  come  exclusively  from 
the  manufactory  of  the  one  who  offers  it  for  sale.  It  shall  be  produced 
by  grinding  scorified  rocks,  obtained  by  calcining  to  the  point  of  vitrifi- 
cation, of  an  intimate  mixture  of  carbonate  of  lime  and  clay,  carefully 
mixed,  and  chemically  and  physically  homogeneous  throughout. 

ART.  2.  The  administration  reserves  the  right,  under  conditions 
which  it  determines,  to  supervise  the  manufacture,  the  storing  at  the 
factory,  and  the  shipping  of  the  cement. 

For  this  purpose  the  engineer  or  his  representative  shall  have  access 
at  all  times  to  all  parts  of  the  factory  concerned  ;  and  he  may — 

i°.  Do  whatever  he  thinks  necessary  to  make  sure  of  the  composi- 
tion of  the  crude  pastes  used. 

2°.   Supervise  the  sorting  after  calcining. 

3°.  Follow  the  cement  after  the  sorting  to  the  special  cases  where  it 
is  to  be  stored  after  grinding. 

4°.  Supervise  the  packing  when  it  is  taken  from  the  cases,  and  also 
the  shipping  of  the  cement. 

5°.  Place  special  agents  permanently  at  the  factory  for  the  above- 
stated  purposes. 

ART.  4.  Every  partial  lot  of  cement,  on  its  arrival  at  the  storehouse 
of  the  works,  must  be  examined  as  to  dryness.  No  bag  shall  be  allowed 
to  enter  which  has  been  exposed  to  dampness,  or  whose  contents  is  not 
entirely  pulverulent  throughout.  Then  the  part  allowed  to  enter,  as  far 
as  dryness  is  concerned,  shall  be  submitted  to  the  tests  prescribed  for, 
1°,  density;  2°,  chemical  composition  ;  3°,  time  of  setting;  4°,  absence 
of  cracks  after  setting;  5°,  strength  of  briquettes  of  neat  cement;  6°, 
strength  of  briquettes  of  cement  with  normal  sand. 

The  engineer,  or  his  representative,  shall  take  some  cement  from  one 
or  more  bags  chosen  arbitrarily  at  such  points  as  he  shall  decide,  bin 
without  mixing  cement  from  different  bags.  He  shall  then  proceed  to 
the  tests,  observing  the  precautions  prescribed.  Each  of  the  specimens 
thus  chosen  must  satisfy  separately  the  conditions  prescribed  ;  the 
measures  to  be  taken  in  regard  to  the  whole  of  a  partial  lot  being  those 
suitable  for  the  specimen  giving  the  least  satisfactory  result. 

ART.  5  gives  very  elaborate  instructions  in  regard  to  the  determination 
of  the  minimum  weight  per  litre  of  cement  that  has  passed  a  sieve  of  5000 
meshes  per  square  centimetre.  A  portion  of  this  article  is  as  follows. 
viz.: — To  obtain,  under  conditions  always  comparable,  an  unheapecl  litre 


7 26  APPLIED   MECHANICS. 


of  the  fine  dust,  produced  by  sifting  cement  through  a  sieve  of  5000 
meshes  per  square  centimetre,  we  place  on  a  firm  support  a  measure  of 
one  litre  capacity  ;  above  this  measure  we  arrange  a  plane  inclined  at 
45°,  formed  of  a  sheet  of  zinc  50  cm.  long,  whose  horizontal  lower  edge 
shall  be  fixed  one  centimetre  above  the  level  of  the  upper  plane  of  the" 
measure;  we  pour,  gently,  the  cement  dust,  by  means  of  a  spoon,  onto 
the  inclined  plane  at  the  top,  until  the  measure  is  a  little  more  than 
filled,  and  we  remove  the  excess  of  cement  by  sliding  over  the  edges  of 
the  measure  a  straight-edge  held  in  a  vertical  plane.  During  this  entire 
operation  the  measure  m'ust  not  be  subjected  to  jar  or  shock.  To  obtain 
the  weight  of  a  litre,  we  make  one  single  weighing  of  the  total  amount 
of  five  measures,  filled  with  the  above-described  precautions. 

ART.  6.  Every  cement  in  which  the  chemical  analysis  shall  show 
more  than  \%  sulphuric  acid  or  compounds  of  sulphur  in  measurable 
proportion  shall  be  rejected. 

ART.  7.  All  cement  will  be  declared  suspected  in  which  chemical 
analysis  shows  more  than  4%  of  oxide  of  iron,  or  which  has  a  value  less 
than  -^  for  the  ratio  between  the  total  weight  of  the  combined  silicon 
and  aluminum,  on  the  one  hand,  and  the  lime  on  the  other. 

ART.  8.  In  the  tests  of  neat  cement,  the  cement  shall  be  mixed  in 
sea-water.  The  water  and  air  during  mixing  shall  be  kept  as  nearly  as 
possible  between  15°  and  18°  C. 

To  determine  the  proper  proportion  of  water  to  mix  with  the  cement 
we  make  the  following  preliminary  test : 

The  mortar  is  obtained  by  taking  900  grammes  of  cement,  and  pour- 
ing on  it  all  at  once  the  water  to  be  used,  mixing  the  mortar  with  a 
trowel  on  a  marble  slab  for  five  minutes  from  the  moment  of  pouring 
the  water. 

The  quantity  of  water  used  shall  be  considered  normal  if  the  mortar 
forms  a  firm  paste,  well  united,  brilliant  and  plastic,  satisfying  the  follow- 
ing conditions : 

i°.  The  consistency  of  the  paste  must  not  change  if  the  mixing  goes 
on  for  eight  minutes  instead  of  five. 

2°.  A  small  quantity  of  paste  taken  with  the  trowel  and  let  fall  on  the 
marble  from  about  50  cm.  must  detach  itself  from  the  trowel  without 
leaving  any  adhering  to  the  trowel,  and,  after  its  fall,  it  must  preserve 
approximately  its  form  without  cracking. 

3°.  A  small  quantity  of  paste  being  taken  in  the  hand,  it  must  be 
sufficient  to  give  it  some  light  taps  to  give  it  a  rounded  form  and  to 
make  the  water  come  to  the  surface ;  it  must  neither  flatten  out  com- 


HYDRAULIC  CEMENTS  AND    BRICK  PIERS. 


pletely  nor  stick  to  the  skin,  and  if  the  ball  be  let  fall  from  half  a  metre,  it 
must  preserve  a  rounded  form  (slightly  flattened),  without  cracks. 

4°.  With  less  water  the  paste  should  be  dry,  not  well  united,  and 
should  show  cracks  in  falling.  With  more  water  it  should  have  a  muddy 
consistency,  with  adherence  to  the  trowel. 

After  making  a  series  of  successive  approximations  we  must  adopt  as 
normal  proportion  the  greatest  proportion  of  water  tried  which  shall 
have  produced  a  plastic  and  not  a  muddy  paste  satisfying  the  conditions 
stated. 

ART.  9.  A  part  of  this  article  reads  as  follows :  With  a  part  of  the 
paste  thus  obtained  we  fill  a  cylindrical  metal  box  of  0.04  m.  height  and 
0.08  m.  diameter,  jarring  it  a  few  seconds,  and  leaving  the  water  that 
rises  to  the  top.  Then  suspend,  by  a  cord  passing  over  a  pulley,  a  Vicat 
needle  of  300  grams  weight  and  a  square  section  i  mm.  on  a  side,  and 
lower  it  gradually. 

The  beginning  of  the  set  is  taken  as  the  time  when  the  needle  ceases 
to  penetrate  to  the  bottom  of  the  mould,  and  the  end  of  the  set,  as  the 
time  when  the  needle  no  longer  penetrates  appreciably. 

Times  are  estimated  from  the  moment  when  the  water  is  poured  on 
the  dry  powder. 

If  the  cement  begins  to  set  before  thirty  minutes  or  completes  its  set 
before  three  hours,  the  partial  lot  shall  be  rejected  ;  the  temperature 
during  the  operation  having  been  between  15°  and  18°  C. 

ART.  10  prescribes  a  form  of  test  to  guard  against  the  presence  of 
cracks  after  setting. 

ART.  ii.  The  paste  for  tensile  tests  of  neat  cement  is  obtained  by 
mixing  with  a  trowel,  on  a  marble  slab,  during  5  minutes,  900  grammes 
of  cement  with  the  normal  quantity  of  water,  as  already  determined. 
Each  mixing  will  furnish  paste  for  6  briquettes.  Make  three  successive 
mixings  to  obtain  18  briquettes,  which  is  the  number  to  be  used  in  each 
test. 

The  form  of  the  briquette  is  prescribed,  the  thickness  being  om.o222, 
the  smallest  section  being  om.O225  wide ;  area,  5  square  centimetres. 

Put  the  moulds  on  a  marble  slab,  and  fill  eich  set  of  six  with  one 
mixing,  putting  enough  in  each  mould  at  once  so  that  it  shall  overflow. 
Pack  with  the  flat  of  the  trowel.  When  the  filling  is  complete,  give 
little  taps  with  the  trowel  handle  on  the  side  to  disengage  bubbles 
of  air. 

As  soon  as  the  consistency  of  the  cement  permits,  smooth  off  the 
upper  surface  even  with  the  mould  by  using  the  blade  of  a  knife. 


728  APPLIED    MECHANICS. 

After  the  cement  has  set  remove  the  moulds,  leaving  the  briquettes 

on  the  slab. 

During  the  first  24  hours  the  briquettes  must  be  kept  on  the  slab,  in 
a  damp  atmosphere,  free  from  currents  of  air  and  the  direct  rays  of  the 
sun,  at  a  temperature  of  from  15°  to  18°  C. 

After  24  hours  immerse  them  in  sea-water,  the  water  to  be  renewed 
every  week,  and  kept,  as  nearly  as  possible,  at  a  temperature  between 
1 5°  and  18°  C. 

For  each  sample  of  cement  to  be  tested  make  18  briquettes  of  neat 
cement,  of  which  6  are  to  be  broken  7  days  from  the  time  of  mixing,  6 
at  the  end  of  28  days,  and  6  at  the  end  of  84  days.  For  each  series  take 
one  briquette  from  each  mixing. 

The  testing-machine  prescribed  is  one  where  the  tension  is  obtained 
by  pouring  a  jet  of  grains  of  lead  into  a  vase  at  the  end  of  a  second 
lever.  Among  the  six  results  in  each  series,  choose  the  three  highest ; 
the  mean  of  these  three  shall  be  considered  to  be  the  strength  of  the 
sample  tested  at  that  time. 

ART.  12.  b.  The  resistance  of  briquettes  of  neat  cement  at  the  end  of 
the  7th  day  must  be  at  least  20  kilogrammes  per  square  centimetre.  It 
must  be  at  least  35  kg.  at  the  end  of  the  28th  day.  Every  partial  lot 
whence  comes  a  sample  not  satisfying  these  two  conditions  shall  be 
rejected. 

ART.  13.  The  strength  per  square  cm.  at  the  end  of  28  days  must  be 
at  least  5  kg.  greater  than  that  at  the  end  of  7  days;  otherwise  the 
partial  lot  shall  be  suspected,  the  suspicion  not  to  be  removed  unless  the 
strength  at  the  end  of  28  days  is  at  least  55  kg. 

ART.  14.  The  strength  per  square  cm.  at  the  end  of  84  days  must  be 
at  least  45  kg.  It  must  also  exceed  the  strength  at  the  end  of  28  days 
when  the  latter  was  not  at  least  55  kg.  Every  partial  lot  not  satisfying 
these  conditions  to  be  rejected. 

The  tests  of  cement  mortar  are  made  on  briquettes  of  mortar  com- 
posed of  one  part  by  weight  of  cement  to  three  of  normal  sand,  the 
latter  being  furnished  by  the  Administration,  and  being  such  as  will  pass 
through  a  sieve  of  64  meshes  per  square  centimetre  and  be  rejected  by 
one  of  144  meshes  per  square  centimetre. 

The  amount  of  water  used  is  12%  of  the  total  weight  of  cement  and 
sand. 

Very  minute  directions  are  given  in  regard  to  the  mixing  and  pre- 
paring the  briquettes  very  similar  to  those  for  neat  cement,  and  then 
the  specifications  proceed  as  follows,  viz.:  For  each  sample  of  cement 


HYDRAULIC    CEMENTS  AND    BRICK  PIERS.  729 

we  make  18  briquettes  of  normal  sand  mortar,  of  which  6  are  to  be 
broken  at  the  end  of  7  days,  6  at  the  end  of  28  days,  and  6  at  the  end  of 
84  days ;  using  in  each  series  a  briquette  from  each  of  the  six  different 
mixtures  in  which  the  mortar  is  to  be  made.  Of  the  six  results  in  each 
series  we  take  the  three  highest,  and  the  mean  of  these  is  the  figure  ad- 
mitted iOr  the  resistance  wf  ..ne  -nortar. 

ART.  17.  c.  The  strength  of  normal  sand  mortar  at  the  end  of  seven 
days  must  be  at  least  8  kg.  per  sq.  cm.,  and  at  the  end  of  28  days  at 
least  15  kg.  per  sq.  cm.  Each  partial  lot  whence  comes  a  sample  not 
satisfying  these  conditions  is  to  be  rejected. 

ART.  1 8.  The  resistance  at  the  end  of  28  days  must  exceed  that  at 
the  end  of  7  days  by  at  least  2  kilogrammes,  otherwise  the  partial  lot  is 
to  be  suspected. 

ART.  19.  The  resistance  at  the  end  of  84  days  must  be  at  least  18 
kilogrammes,  and  it  must  exceed  the  resistance  at  the  end  of  28  days. 
Every  partial  lot  whence  comes  a  sample  not  satisfying  these  conditions 
should  be  rejected. 

The  German,  the  Swiss,  and  other  specifications  may  be 
found  in  Candlot's  book;  but  a  portion  of  those  of  the  American 
Society  for  Testing  Materials  will  be  quoted  here. 

AMERICAN  SOCIETY  FOR  TESTING  MATERIALS. 

REPORT   OF   COMMITTEE   ON   STANDARD  SPECIFICATIONS  FOR 

CEMENT. 

GENERAL  OBSERVATIONS. 

1.  These  remarks  have  been  prepared  with  a  view  of  pointing  out 
the  pertinent  features  of  the  various  requirements  and  the  precautions 
to  be  observed  in  the  interpretation  of  the  results  of  the  tests. 

2.  The  Committee  would  suggest  that  the  acceptance  or  rejection 
under  these  specifications  be  based  on  tests  made  by  an  experienced 
person  having  the  proper  means  for  making  the  tests. 

3.  Specific  Gravity. — Specific  gravity  is  useful  in  detecting  adultera- 
tion or  underburning.     The  results  of  tests  of  specific  gravity  are  not 
necessarily  conclusive  as  an  indication  of  the  quality  of  a  cement,  but 
when  in  combination  with  the  results  of  other  tests  may  afford  valuable 
indications. 

4.  Fineness. — The  sieves  should  be  kept  thoroughly  dry. 

5.  Time  oj  Setting. — Great  care  should  be  exercised  to  maintain 
the  test  pieces  under  as  uniform  conditions  as  possible.     A  sudden 


73°  APPLIED    MECHANICS. 

change  or  wide  range  of  temperature  in  the  room  in  which  the  tests  are 
made,  a  very  dry  or  humid  atmosphere,  and  other  irregularities  vitally 
affect  the  rate  of  setting. 

6.  Tensile  Strength. — Each  consumer  must  fix  the  minimum  re- 
quirements for  tensile  strength  to  suit  his  own  conditions.     They  shall, 
however,  be  within  the  limits  stated. 

7.  Constancy  of  Volume. — The  tests  for  constancy  of  volume  are 
divided  into  two  classes,  the  first  normal,  the  second  accelerated.     The 
latter  should  be  regarded  as  a  precautionary  test  only,  and  not  infallible. 
So  many  conditions  enter  into  the  making  and  interpreting  of  it  that 
it  should  be  used  with  extreme  care. 

8.  In  making  the  pats  the  greatest  care  should  be  exercised  to  avoid 
initial  strains  due  to  molding  or  to  too  rapid  drying-out  during  the  first 
twenty-four  hours.     The  pats  should  be  preserved  under  the  most 
uniform  conditions  possible,  and  rapid  changes  of  temperature  should 
be  avoided. 

9.  The  failure  to  meet  the  requirements  of   the  accelerated  tests 
need  not  be  sufficient  cause  for  rejection.     The  cement  may,  however, 
be  held  for  twenty-eight  days,  and  a  retest  made  at  the  end  of  that 
period.     Failure  to  meet  the  requirements  at  this  time  should  be  con- 
sidered sufficient  cause  for  rejection,  although  in  the  present  state  of  our 
knowledge  it  cannot  be  said  that  such  failure  necessarily  indicates 
unsoundness,  nor  can  the  cement  be  considered  entirely  satisfactory 
.simply  because  it  passes  the  test. 

GENERAL  CONDITIONS. 
Of  these  the  first  eight  will  not  be  quoted  here. 

9.  All   tests   made   in   accordance   with  the  methods  proposed  by 
the  Committee  on  Uniform  Tests  of  Cement  of  the  American  Society 
of  Civil  Engineers,  presented  to  the  Society,  January  21,  1903,  and 
amended  January  20,  1904,  with  all  subsequent  amendments  thereto. 
(See  addendum  to  these  specifications.) 

10.  The  acceptance  of  rejection  shall  be  based,  on  the  following 
requirements : 

NATURAL  CEMENT. 

11.  Definition. — This  term  shall  be  applied  to  the  finely  pulverized 
product  resulting  from  the  calcination  of  an  argillaceous  limestone  at  a 
temperature  only  sufficient  to  drive  off  the  carbonic  acid  gas. 


AMERICAN  SOCIETY  FOR    TESTING   MATERIALS.      731 

12.  Specific  Gravity. — The  specific  gravity  of  the  cement  thoroughly 
dried  at  100°  C.,  shall  be  not  less  than  2.8. 

13.  Fineness. — It  shall  leave  by  weight  a  residue  of  not  more  than 
10  per  cent  on  the  No.  100,  and  30  per  cent  on  the  No.  200  sieve. 

14.  Time  of  Setting. — It  shall  develop  initial  set  in  not  less  than  ten 
minutes,  and  hard  set  in  not  less  than  thirty  minutes,  nor  more  than 
three  hours. 

15.  Tensile    Strength. — The    minimum    requirements    for    tensile 
strength  for  briquettes  one  inch  square  in  cross-section  shall  be  within 
the  following  limits,  and  shall  show  no  retrogression  in  strength  within 
the  periods  specified :  * 

Age.                                                Neat  Cement.  Strength. 

24  hours  in  moist  air 50-100  Ibs. 

7  days  (I  day  in  moist  air,     6  days  in  water) 100-200    " 

28     "     (I     "     "       "       "     27      "     "       "    ) 200-300    " 

One  Part  Cement,   Three  Parts  Standard  Sand. 

7  days  (I  day  in  moist  air,     6  days  in  water) 25-  75  Ibs. 

28      "     (I     "     "       "       "     27     "     "       "     )     75-150    *' 

1 6.  Constancy  of   Volume. — Pats   of  neat   cement   of  about   three 
inches  in  diameter,  one-half  inch  thick  at  center,  tapering  to  a  thin 
edge  shall  be  kept  in  moist  air  for  a  period  of  twenty-four  hours. 

(a)  A  pat  is  then  kept  in  air  in  normal  temperature. 

(b)  Another  is  kept  in  water  maintained  as  near  70°  F.  as  prac- 
ticable. 

17.  These  pats  are  observed  at  intervals  for  at  least  28  days,  and, 
to  satisfactorily  pass  the  tests,  should  remain  firm  and  hard  and  show 
no  signs  of  distortion,  checking,  cracking  or  disintegrating. 

PORTLAND  CEMENT. 

1 8.  Definition. — This    term    is    applied    to    the    finely    pulverized 
product  resulting  from  the  calcination  to  incipient  fusion  of  an  intimate 
mixture  of  properly  proportioned  argillaceous  and  calcareous  materials, 
and  to  which  an  addition  no  greater  than  3  per  cent  has  been  made 
subsequent  to  calcination. 

*  For  example  the  minimum  requirements  for  the  24-hour  neat  cement  test 
should  be  some  value  within  the  limits  of  50  and  100  Ibs.,  and  so  on  for  each 
period  stated. 


73 2  APPLIED   MECHANICS. 

19.  Specific   Gravity. — The  specific  gravity  of  the   cement,   thor- 
oughly dried  at  100°  C.,  shall  not  be  less  than  3.10. 

20.  Fineness. — It  shall  leave  by  weight  a  residue  of  not  more  than 
8  per  cent  on  the  No.  100,  and  not  more  than  25  per  cent  on  the  No. 
200  sieve. 

21.  Time  o)  Setting. — It  shall  develop  initial  set  in  not  less  than 
thirty  minutes,  but  must  develop  hard  set  in  not  less  than  one  hour,  nor 
more  than  ten  hours. 

22.  Tensile    Strength. — The    minimum    requirements    for    tensile 
strength  for  briquettes  one  inch  square  in  section  shall  be  within  the 
following  limits,  and  shall  show  no  retrogression  in  strength  within  the 
periods  specified :  * 

Age.  Neat  Cement.  Strength. 

24  hours  in  moist  air 150-200  Ibs. 

7  days  (i  day  in  moist  air,    6  days  in  water) 450-550    " 

28     "     (I    "    "      "      "     27     "     "      "    ) 550-650    " 

One  Part  Cement,  Three  Parts  Standard  Sand. 

7  days  (I  day  in  moist  air,    6  days  in  water) 150-200  Ibs. 

28     "     (I    "    "      "      "     27     "     "      ««    ) 200-300    " 

23.  Constancy  of  Volume. — Pats  of  neat  cement  about  three  inches 
in  diameter,  one-half  inch  thick  at  the  center,  and  tapering  to  a  thin 
edge,  shall  be  kept  in  moist  air  for  a  period  of  twenty-four  hours. 

(a)  A  pat  is  then  kept  in  air  in  normal  temperature  and  observed 
at  intervals  for  at  least  twenty-eight  days. 

(b)  Another  pat  is  kept  in  water  maintained  as  near  70°  F.  as 
practicable,  and  observed  at  intervals  for  at  least  twenty-eight  days. 

(c)  A  third  pat  is  exposed  in  any  convenient  way  in  an  atmosphere 
of  steam,  above  boiling  water,  in  a  loosely  closed  vessel  for  five  hours. 

24.  These  pats,  to  satisfactorily  pass  the  requirements,  shall  remain 
firm  and  hard  and  show  no  signs  of  distortion,  checking,  cracking  or 
disintegrating. 

*  For  example  the  minimum  requirement  for  24-hour  neat  cement  test  should 
be  some  value  within  the  limits  of  150  and  200  Ibs.,  and  soon  for  each  period 
stated. 


HYDRAULIC   CEMENTS  AND    BRICK  PIERS.  733 


25.  Sulphuric  Acid  and  Magnesia. — The  cement  shall  not  contain 
more  than  1.75  per  cent  of  anhydrous  sulphuric  acid  (SOs),  nor  more 
than  4  per  cent  of  magnesia  (MgO). 

Bauschinger  has  also  made  a  large  number  of  compression 
tests,  accounts  of  which  may  be  found  in  Hefte  i,  7,  and  8  of  the 
Mittheilungen,  but  for  these  the  student  is  referred  to  the  Mittheil- 
ungen. 

PRECAUTIONS   TO   BE   OBSERVED   IN  TESTING   CEMENTS. 

The  results  obtained  by  testing  different  samples  of  the  same 
cement  will  vary  with — 

i°.  The  percentage  of  water  used  in  mixing. 

2°.  The  length-  of  time  the  sample  has  been  kept  under  water 
and  also  the  length  of  time  it  has  been  kept  in  the  air  before 
testing. 

3°.  The  temperature  of  the  water  with  which  it  was  mixed, 
and  also  of  that  in  which  it  was  kept ;  also  the  temperature  of  the 
air  in  which  it  was  kept. 

4°.  The  rapidity  of  breaking. 

Hence,  in  order  that  our  results  may  be  of  value,  we  must 
take  pains  to  regulate  all  these  matters. 

But  another  and  all-important  matter  that  has  not  received  the 
necessary  amount  of  attention  is,  that  some  means  should  be 
adopted  for  distributing  the  pull,  in  the  case  of  a  tension  test, 
evenly  over  the  section  of  the  briquette,  and  in  the  case  of  a  com- 
pression test  for  distributing  the  thrust  evenly  over  the  surface 
of  the  specimen.  In  the  ordinary  cement- testing  machines  to 
be  found  in  the  market  there  is  generally  no  adequate  provision 
for  this  purpose,  and  this  is  the  reason  why  so  great  a  variation 
exists  in  the  results  obtained  with  the  same  cement  by  so  many 
experimenters.  For  a  fuller  account  of  this  matter  see  Trans. 
Am.  Soc.  Mech.  Engrs.  for  1888,  page  172. 

The  following  is  a  summary  of  a  part  of  a  paper  read  by  Mr. 


734  APPLIED    MECHANICS. 

James  E.  Howard  of  Watertown  Arsenal,  before  the  Assoc.  of 
Am.  Portland  Cement  Mfr.,  in  April,  1905.  He  says: 

i°.  That  a  loo-mesh  sieve  has  openings  o."oo58  diam. 

That  a  200-mesh  sieve  has  openings  o."oc>3i  diam. 

That  a  No.  20  bolting  cloth  has  openings  o."oo27  diam. 

He  advises  the  use  of  the  latter  for  the  separation  of  the  fine 
from  the  coarse  particles. 

2°.  That  while  he  obtained  for  freshly  ground  Portlands 
specific  gravities  in  the  vicinity  of  3.1,  there  were  a  number  of 
natural  cements  examined,  which  had  substantially  the  same 
values  as  Portlands,  although  some  brands  fell  below  3. 

That  hydration,  partial  or  complete,  lowers  the  specific  gravity. 
That  hydration  begins  at  once,  goes  on  more  quickly  in  the  finer 
particles  and  more  slowly  in  the  coarser  ones.  That,  in  some 
cases,  hydration  was  not  complete  at  the  end  of  five  days.  Hence, 
that  the  usual  arbitrary  methods  of  determining  the  beginning 
and  the  end  of  the  set  do  not  show  the  beginning  and  end  of 
hydration. 

That,  as  a  rule,  the  compressive  strength  of  the  cement  will  not 
be  diminished,  if  a  period  of  about  eight  hours  intervene  between 
gauging  and  use. 

3°.  That  exposure  to  high  temperatures  is  liable  to  lead  to 
ultimate  disintegration. 

4°.  That  a  number  of  compression  specimens  were  moulded 
under  pressures  varying  from  7000  to  14000  pounds  per  square 
inch,  continued  for  40  hours  or  more,  and  were  subsequently 
tested,  at  ages  of  i  and  2  months.  They  developed  phenomenal 
strength,  a  sample  of  neat  cement  showing  a  strength  of  22050 
pounds  per  square  inch  at  the  age  of  57  days,  and  i  to  i  mortar 
19120  pounds  per  square  inch  at  the  age  of  i  month. 

Setting  under  high  pressures  admits  of  the  use  of  smaller 
quantities  of  water  in  gauging;  in  one  case  only  5  per  cent  having 
been  used. 


HYDRAULIC  CEMENTS  AND    BRICK  PIERS.  735 


TESTS   OF    FULL-SIZE   PIECES. 

Inasmuch  as  cement  and  mortar  are  almost  always  used  as 
binding  materials,  tests  of  full-size  pieces  in  which  they  enter  are 
those  of  some  form  of  masonry.  Of  such  tests  the  number  is  not 
large,  and  those  that  will  be  quoted  here  are  some  tests  of  reen- 
forced  concrete,  and  some  of  brick  pieces. 

Concrete  is  composed  of  mortar  and  some  hard  material,  as 
gravel,  broken  stone,  cinder,  etc.,  the  general  plan  being  to  so 
proportion  them  that  the  cement  shall  approximately  fill  the  voids 
in  the  sand,  and  that  the  mortar  shall  approximately  fill  the  voids 
in  the  broken  stone,  or  other  hard  material  used. 

Reenforced  concrete,  which  is  made  by  imbedding  in  the 
concrete,  iron  or  steel  bars,  wire  mesh  or  expanded  metal,  etc., 
is  now  attracting  a  great  deal  of  attention. 

COLUMNS. 

An  extensive  series  of  tests  of  columns  of  reenforced  concrete 
is  now  being  carried  on  at  the  Watertown  Arsenal,  and  the  follow- 
ing table,  which  gives  a  summary  of  the  tests  of  this  kind  already 
published,  is  quoted  from  "Tests  of  Metals  for  1904." 


APPLIED    MECHANICS. 


§1 
il 


o^  I 

Wffi  .§ 

f-<  H  ^ 

M^  I 


SSOJ£)  UO 


3(J  'scn 

jo  ItiSp 


ooooooooooooooooooo 

OOOOOOOOOOOOOOt-OOwvOO 

O    H  00  00    IN    M    «5  OOO    t^  IN  OC    OOO  OC 


•SSOJQ 


HYDRAULIC   CEMENTS  AND   BRICK  PIERS. 


737 


The  following  table  gives  the  results  of  a  set  of  tests  made  in 
the  Laboratory  of  Applied  Mechanics  of  the  Mass.  Institute  of 
Technology,  upon  reenforced  concrete  columns,  the  concrete 
consisting  of  i  part  Portland  cement  (Star  brand),  2  parts  sand, 
and  6  parts  trap-rock. 


, 

m 

« 

1 

n 

e 

-e 

5 

_c 

.a  § 

W)  u 

1 

-1 

4 

M 

Remarks. 

^ 

•1 

£ 

J 

0 

CO 

E 

W^ 

In.  In. 

i 

30 

8X8 

17 

I 

I 

P 

107000 

Crushed  at  end. 

2 

30 

*  • 

17 

I 

I 

T 

127000 

Buckled,  then  crushed  at  end. 

3 

29 

1  ' 

12 

I 

I 

P 

IOOOOO 

Buckled,  then  crushed  at  end. 

4 

28 

4  • 

12 

I 

I 

T 

I  26000 

Crushed  at  end.   Poorly  made.    Crushed 

portion  cut  off;  the  rest  bore  150000 

Ibs.  at  40  days. 

5 

32 

" 

6 

I 

I 

P 

138000 

Crushed    at   middle,  then    sheared    off 

along  rod  to  end. 

6 

31 

" 

6 

I 

I 

T 

133000 

Crushed  at  end. 

7 

•  ' 

*7 

I 

P 

136000 

Crushed  at  end,  shearing  obliquely. 

8 

25 

•  « 

17 

I 

T 

154000 

Crushed  at  end,  breaking  off  3  feet. 

9 

35 

'  ' 

4 

P 

182000 

Crushed  at  end. 

10 

34 

*  ' 

17 

4 

T 

167000 

Crushed  at  end,  concrete  rather  poor 

and  rough  at  that  end. 

ii 

3i 

'  ' 

12 

4 

j. 

T 

147000 

Crushed  and  split  open  at  end. 

I  2 

32 

4  • 

I  2 

4 

f 

P 

153000 

Crushed  and  split  open  at  end. 

13 

29 

'  ' 

6 

4 

T 

158000 

Crushed  and  split  open  at  end. 

14 

4  4 

6 

4 

P 

244000 

Crushed  at  end. 

IS 

35 

10  X  10 

17 

P 

215000 

Broke  off  clean  for  3  or  4  feet  at  end. 

16 

35 

*  4 

6 

P 

240000 

Sheared  diagonally  at  end. 

*7 

45 

4  4 

6 

T 

228400 

Sheared  diagonally  at  end,  and  broke 

back  for  half  the  length. 

18 

3i 

4  ' 

12 

i 

T 

262000 

Sheared  diagonally  near  end. 

19 

29 

4  ' 

12 

P 

257000 

Crushed  at  end. 

20 

28 

4  ' 

12 

4 

T 

300000 

Not  broken. 

21 

29 

*  4 

12 

4 

P 

274000 

Crushed  at  end.     Wedge-shaped  piece 

forced  in  between  rods. 

REENFORCED  CONCRETE  BEAMS. 

While  more  or  less  theorizing  has  been  done  by  different 
people  regarding  suitable  formulae  for  use  in  the  case  of  reen- 
forced concrete  beams,  the  writer  believes  that  more  tests  are 
needed  before  such  theorizing  can  be  placed  upon  a  permanent 
basis.  Some  results  of  tests  of  full-size  reenforced  concrete 
beams  are  given  in  the  following  tables: 


738 


APPLIED    MECHANICS. 


SOME  TESTS  MADE  IN  THE  LABORATORY  OP  APPLIED  MECHANICS  OP  THE 
MASSACHUSETTS   INSTITUTE   OF   TECHNOLOGY. 

Concrete  was  of  the  same  composition  as  in  the  case  of  the  columns.     Size  of  beams 
8"Xi2".     Span  n'.     When  load  was  at  two  points  they  were  44"  apart,  and  symmet- 


No.  of 
Beam. 

Age  in 
Days. 

No.,  Size,  and  Kind  of 
Bars  near  Bottom. 

Manner  of 
Loading 
at  Time  of 
Fracture. 

Weight 
of 
Beams 
in  Lbs. 

Breaking 
Load, 
Exclusive 
of 
Weight 
of 
Beam, 
in  Lbs. 

Maximum 
Bending- 
Moment 
at 
Fracture 
in 
In.-lbs. 

No. 

Side  of, 
in 
Sq.  Ins. 

Plain,  P, 
or 
Twisted, 
T. 

I 

40 

Center 

1198 

1302 

62733 

2 

40 

'i 

T 

" 

1  200 

1300 

62700 

3 

39 

T 

At  two  points 

1205 

10095 

241973 

4 

38 

1 

T 

Center 

1160 

13680 

470580 

5 

50 

I 

T 

<  < 

1290 

14710 

5°67i5 

6 

5° 

T 

lt 

1204 

I5796 

54H34 

7 

41 

i 

4 

T 

" 

"95 

12805 

442283 

8 

41 

2 

T 

" 

1240 

18760 

639540 

9 

42 

2 

T 

" 

1274 

23105 

783486 

10* 

42 

2 

T 

ft 

1279 

21105 

717569 

nt 

45 

2 

T 

" 

1294 

23!°5 

783816 

12 

30 

2 

;; 

T 

At  two  points 

1282 

24200 

553553 

J3 

3i 

2 

1 

4 

T 

1292 

29200 

663718 

I4t 

30 

2 

i 

T 

i34i 

24200 

554527 

15 

53 

I 

I 

P 

1292 

I525° 

356818 

16 

49 

I 

I 

T 

I2II 

16500 

382982 

17 

43 

2 

f 

P 

1271 

1595° 

370700 

18 

40 

2 

f 

T 

I20O 

19600 

438807 

*9 

35 

4 

* 

P 

I26l 

17500 

378065 

20 

33 

4 

i 

T 

1213 

2OOOO 

433329 

21 

57 

I 

^ 

P 

1213 

12500 

295OI5 

22 

54 

2 

|. 

T 

1248 

2«2250 

510092 

23 

57 

2 

I 

P 

1221 

2O250 

465647 

24 

47 

4 

f 

T 

1203 

19250 

44335° 

25 

5° 

4 

f 

P 

1192 

15250 

355168 

26 

40 

2 

1 

T 

1215 

24^50 

553548 

27 

49 

I 

I 

T 

1222 

21750 

498688 

*  Also  one  bar  V  square  near  top. 
t  Also  two  bars  i"  square  near  top. 

In  beams  Nos.  12  and  13  there  were,  on  each  side  of  the  middle  of  the  span, 
eight  pieces  of  J-inch  twisted  steel  wire,  bent  in  the  form  of  a  U  enclosing  the  two 
reenforcing  rods.  In  No.  13  they  were  vertical,  and  in  No.  12  they  were  inclined 
at  45°  to  the  horizon,  sloping  upward  away  from  the  middle.  In  No.  14  the 
wire  pieces  were  in  the  form  of  a  square,  vertical,  and  enclosing  all  four  of  the 
bars.  Nos.  26  and  27,  each  contained,  in  addition  to  the  bars,  a  vertical  layer 
of  expanded  metal,  extending  throughout  their  length  and  height. 


HYDRAULIC  CEMENTS  AND   BRICK  PIERS. 


739 


SOME  TESTS  MADE  AT  THE   ENGINEERING   EXPERIMENT    STATION, 
UNIVERSITY  OF   ILLINOIS. 

PLAIN  CONCRETE. 

Beams  were  12"  wide  by  13$"  deep.     Mixture  by  volume  was  i  part  Chicago  A  A 
Portland  cement;   3  parts  clean,  sharp  sand;   6  parts  broken  limestone  (i"-i $-.")• 


Beam  No. 

Length. 

Age,  days. 

Span. 

Maximum 
Applied  Load. 

Modulus 
of  Rupture. 

Ft.  Ins. 

Ft.  Ins. 

8 

J5   4 

64 

14 

3600 

412 

ii 

15   4 

65 

14 

2600 

337 

18 

JS   4 

64 

14 

2400 

322 

26 

12 

62 

10   8 

55°° 

39° 

3° 

12 

62 

10   8 

4800 

355 

23 

9   6 

61 

8   6 

6355 

347 

31 

9   6 

62 

8   6 

8000 

422 

24 

6 

61 

5 

10240 

2  99 

25 

6 

64 

5 

10200 

299 

REENFORCED  CONCRETE. 

Size  of  beams:  Length,  15'  4",  Span  14',  breadth  12",  depth  13!",  center  of  metal 
reenforcement  12"  below  top  surface  of  beam.  Loads  applied  at  the  one-third  points 
of  beam. 


Amount  and  Kind 
of  Reenforcement. 

Area  of 
Metal, 
Sq.  Ins. 

Maximum 
Load, 
Lbs. 

3   J"  plain  round 

•59 

9000 

3  i"    "       " 

•59 

9200 

3  \"     "     square 

•75 

9900 

3  i 

//     <  «         « 

•75 

1  0000 

4  i 

//     <  <         <  « 

2.25 

26900 

3 

"  Ransome 

•75 

22800 

3  5 

"  Thatcher 

i.  20 

18400 

3  i 

/^        « 

I  .20 

16600 

3  i 

;"  Kahn 

2.40 

24400 

5 

ft      <  t 

2  .OO 

23000 

4 

ft      <  < 

I.  60 

17200 

3 

y      <  < 

1.  2O 

15000 

6  I"  Johnson 

2.19 

34300 

7y        •• 

5  r    " 

I.4O 
1.  00 

29000 
20900 

5  i"        " 

I.OO 

20600 

3  r    " 

.60 

14000 

3  V        " 

.60 

14000 

BRICKS   AND   BRICK  PIERS. 

In  this  connection  two 
sets  of  tests  of  brick  piers, 
made  at  the  Watertown  Ar- 
senal, will  be  quoted  here. 
The  first  is  taken  from 
Tests  of  Metals  for  1886, 
and  Ihe  second  from  Tests 
of  Metals  for  1904. 

The  tabulation  of  the 
second  series,  which  com- 
prises 26  piers,  is  given  in 
the  table  on  page  742. 


740 


APPLIED    MECHANICS. 


The  first  series  comprises  53  piers,  in  the  construction  of  which  two  kinds  of 
brick  were  used,  viz.,  common  hard-burned  bricks  and  face-bricks,  laid  on  bed, 
with  joints  broken  every  course.  The  tabulation  follows: 

FACE-BRICK   PIKRS. 


Weight 
per 
Cubic 
Foot. 

Actual  Dimensions. 

Sectional 
Area. 

Ultimate  Strength. 

Height. 

Cross-section. 

Total. 

Per 

Sq.  In. 

Per 

Sq.  Ft. 

Ibs. 

ft.       in. 

in. 

in. 

sq.  in. 

Ibs. 

Ibs. 

tons. 

132.7 

2     0. 

7.63 

7.6l 

58.06 

141000 

2428 

174-81 

134-4 

2      0.27 

7.64 

7.63 

58.29 

123400 

2117 

152.42 

130.2 

3  n-95 

7-75 

7.68 

59-52 

I22OI6 

2050 

147  60 

129.7 

4    o. 

7-75 

7.70 

59-68 

116000 

1944 

139-97 

127.6 

6    0.37 

7-75 

7-75 

60.06 

II7II7 

1950 

140.4 

I2Q.6 

6    0.26 

7.85 

7-75 

60.84 

106470 

1750 

126.0 

125.2 

8    0.56 

7.78 

7-75 

60.30 

IO2OOO 

1691 

121.75 

.... 

9  11.27 

7.80 

7.70 

60.06 

100749 

1677 

120.77 

126.8 

10    0.37 

7.82 

7.80 

61.00 

II0500 

1811 

130.39 

126.4 

*5  u.68 

1  1.  60 

ii.  60 

I34-56 

257300 

1912 

137-66 

129.0 

5  11.27 

11-55 

11.50 

132.82 

258IOO 

1943 

139.89 

130.3 

5  ii. 

j  ii-55 
\    4.20 

11.50  \ 
4.10  J 

115.61 

219659 

1900 

136.8 

129.5 

5  10.09 

15-45 

15.40 

237-93 

499653 

.2100 

151.2 

J26.0 

5  11.81 

15-45 

15-35 

237.16 

468700 

1976 

142.27 

COMMON-BRICK  PIERS. 


120.8 

I  10.87 

7-65 

7-52 

57-53 

161000 

2798 

201.45 

123.3 

i  11.13 

7.65 

7.60 

58.14 

157800 

2714 

195.40 

125.4 

4  0.37 

7.60 

7-55 

57-38 

111891 

1950 

140.40 

124.6 

3  11.62 

7.68 

7-58 

58.21 

101867 

1750 

I26.OO 

121.5 

6  1.62 

7-65 

7.60 

58-14 

144300 

2481 

178.63 

123-3 

6  1.18 

7.60 

7.58 

57-6i 

132503 

2300 

165.60 

121.4 

8  1.50 

7-65 

7.63 

58-37 

90474 

1550 

III.60 

121.4 

7  11.98 

7.60 

7-55- 

57-38 

90800 

1582 

H3.90 

121.  0 

10  0.93 

7.60 

7-55 

57.38 

86070 

1500 

108.00 

I23.I 

10  1.31 

7.60 

7-55 

57-38 

104200 

1815 

130.68 

123-5 

i  i  i.  06 

ii.  60 

11.40 

132.24 

307800 

2327 

167.54 

125.8 

i  10.75 

11.38 

ii-35 

129.16 

318500 

2466 

177-55 

124-9 

3  11-58 

H.55 

11.50 

132.83 

224100 

1687 

121.46 

I25.I 

3  ii.  81 

11.40 

11.30 

128.82 

251199 

1950 

140.40 

123.2 

6  0.75 

n-45 

".45 

131.10 

222870 

I7OO 

I22.4O 

121.  7 

6  0.75 

11.48 

11-45 

I3I-45 

216200 

1644 

118.36 

121.  6 

8  1-37 

H.45 

11.40 

130-53 

190700 

1461 

105.19 

;20.8 

8  0.75 

11.50 

11.40 

131.10 

2IIIOO 

l6lO 

115.92 

"9-5 

10   1.  00 

".55 

II.45 

132-25 

178200 

1347 

96.98 

*  Core  built  of  common  brick. 


HYDRAULIC  CEMENTS  AND   BRICK  PIERS. 


741 


COMMON-BRICK  PIERS — Continued. 


Weight 
per 
Cubic 
Foot. 

Actual  Dimensions. 

Sectional 
Area. 

Ultimate  Strength. 

Height. 

Cross-section. 

Total. 

Per 
Sq.  In. 

Pei 

Sq.  Ft. 

Ibs. 

ft.      in. 

in. 

in. 

sq.  in. 

Ibs. 

Ibs. 

tons. 

126.2 

I    11.00 

(  11.40 
j    4-55 

11.40) 

4-50J 

109.48 

271500 

2480 

178.56 

127.7 

2      1.  12 

j  n-45 
(    4-90 

11.40  I 
4-55  ) 

108.23 

265400 

2452 

176.54 

127.3 

4    0.18 

j  n-35 
(    4-70 

"•35  I 
4.70) 

106.73 

198100 

1856 

I33-63 

127.7 

3  11.98 

(    4-80 

11.40) 
4.70) 

107.97 

215200 

1993 

143-49 

II8.8 

6    0.75 

j  H-35 
]    4-90 

11-35  I 
4.6of 

106.28 

162100 

1525 

110.52 

124-3 

8     3-18 

j  n-50 
1    4.90 

11.50) 

4.80  f 

108.73 

184841 

I7OO 

122.40 

125.1 

10    2.27 

(  11.50 

(    4-8o 

1  1  .  40  ) 

4-Sof 

108.06 

157500 

1457 

104.90 

123.0 

6     1.18 

15-50 

15-45 

239.48 

358200 

1495 

107  .  64 

121.  8 

6     1.37 

15.70 

15-65 

245.71 

356600 

104.49 

123.7 

9  11.00 

15.50 

15.40 

238.70 

230200 

964 

69.41 

1  20.  1 

9  11.98 

15-70 

15.60 

244.92 

247400 

IOIO 

72.72 

124.3 

6    0.68 

j  15-45 
}    8.60 

!5-45  I 
8.30  f 

167.32 

270100 

1614 

II6.2I 

125.6 

6    0.68 

j  15.50 
1    8.70 

15-45  ) 
8.6of 

164.66 

260200 

1580 

113.76 

123.4 

9  H.25 

j  15-40 
I    8.40 

15-40  ) 
8.3of 

167.44 

2I2IOO 

1267 

91.22 

128.9 

9  10.56 

j  15-45 
1    8.70 

I5-40  ) 
8.60  f 

163.  II 

202000 

1238 

89.14 

122.3 

*I2      6.5 

1  1.  60 

1  1.  60 

134.56 

2I720O 

1622 

116.78 

125.0 

fl2       6.5 

H-55 

11.40 

131.67 

193300 

1468 

105.69 

117.1 

t  4     o. 

7.90 

7.90 

62.41 

72300 

1158 

83.37 

124.0 

J  3   11-25 

8.00 

8.00 

64.00 

105800 

1654 

119.09 

124.5 

§6     i.i 

(  n-55 

]      4-20 

11-55  j. 
4.20) 

115.76 

60800 

525 

37.80 

*  Laid  with  bond  stones  4  feet  apart. 
$  Face-brick  pier  grouted. 


t  Common-brick  pier  grouted. 

§  Face-brick  pier,  laid  without  mortar. 


.  The  mortar  was  composed  of  Rosendale  cement  i,  sand  2. 
The  piers  were  21  months  old  when  tested.  In  this  series  the 
mortar  was  kept  purposely  the  same  throughout,  so  that  the 
variation  in  strength  should  be  due  to  the  variation  in  dimen- 
sions of  the  piers.  The  mortar,  however,  was  found  to  be 
much  stronger  in  some  places  than  in  others. 

The  tabulation   of  the  second   series,  which  comprises  26 
piers,  is  given  on  page  742. 


742 


APPLIED    MECHANICS. 


-E 

rt 
,3 

Jj 

4^*0"-^  o 

2000 

*Q    N  00                                 l^  >0 

goo                  oo 

£   0   0                          00 

CO 
10 

0 

o 

P 

to  £.2  d 

i4 

rt 

flJjT 

2 

•^   M   N  r^oo  to-  <N  N   cs  o 

^OOMOOMOOM 

OO         OO               OO               OO         OO 

| 

~1H  H, 

^    .0  t^  000    0    M    N    too 

tNt-N^O         OMtOONO         tt-0 

<1) 

*jf§ 

gOOf^OOOOOOCO 

OOtOOO         OOOOOOOO         OOtO 
OO^OOO*         OOMMOO         OOM 

^ 

00-           •      •   0    0 
O   O     •           •     •   O    O 

o        
o        

•d 
§ 

i« 

gtJ  o 

•^r  CO    O\     •                 •   M  O 

«     

£ 

£# 

M«N 

M             

0 

g 

Is 

W    (0 

OTJ   0 
O   C   ° 

O   O   O    O    O   O 

•ooooooooo 

OO-OO-         OO-'OO         OO- 
00-00-         00--00         00- 

O  to    •  oc   M     -         r^o          •  O   O        O   M 

c 
M  -^ 

N     M        .                       .             2     M       .       .                            2     H       . 

«l 

"3 
If 

0*2  o' 

888888888 

yjOOOOOOOOO 

000000         000000         000 

oooooo       oooooo       ooo 
oooooo       oooooo       ooo 

1 

£  3 

M  aSo 

^srssss^affs 

^""S"1    22aS:;§2    §Sa 

0> 

a 

.C     a> 

>  • 

«    .    . 

.OOr-ioOOO-0 

ootootNooo^o-tioo^v: 

C! 
J 

^      M 

•i-5 

a,w^ 

^NMHHHHC.^M 

q   « 

l| 

• 

OdOOOOOOO 
.OroOOOOOOO 

OMOOOOOOOOOOOOCOOO 
OCOOOOOOOMOOOOOOOO 

d. 

>   £ 

0C/2 

1 

^MCNQOO^N    O\O 

J   0   •*  M   i^  ^co   to    C 

<D  t  OO   1000   O   O  10CO   t^  O  O   O>  M  CO  O 

"S 
s 

1   § 

0 

H 

A 

_rt    G 

£   3 

§ 

£   tO   tOO   i>  O    t  v 

o  to  too  otooooo  too  t^  M  <*>  « 

1 

(X  ^ 
"rt    ^ 

c/5.2 

< 

^  

S 

c  "^ 

^fc^ 

8 

Jg  ^  0   0  4  0   0   ^  ^  0 

too'  ^M   tou^o   40   OO^u;^   ^0 

w 

rM 

•"  .-.° 

"^  &0 

•H 

s 

'=  "2 

Q  :  :  :  N  N  :  N/  N  : 

•  •«  :  :  -S8  -^SM  •S'S  •  • 

•» 

g  * 

4 

g  -«««,«,«     itoo 

M    M    10  M  O  O    10  IOO    10  10  IOO      •    tOO  O 

*4J 
§ 

c 

1^       |^|       ^ 

C            C                 C                 C            C 

ew 
* 

1 

1 

I^!|^t^|! 

y||^li|^|l|^l|^i 

00 

X 

.§ 

?lli§llil 

i  ilj  III  silli  Hill 

V 

r\   M   M  r  *   M 

' 

nal  dimensions  12") 

"8 

O  c/ 

;i| 

off 

i 

i  f 

if  ||  i  ill  i||  || 

• 

s 

. 

Tt-  t^OO   t  to  O   O-O   O 

*                * 

£ 

'Z*® 

0) 

too  o  \o  o  o  >oo  o 

to  100   100  OOOOOOOO   100  0  vo 

H 

FUNDAMENTAL   PRINCIPLES.  743 


CHAPTER  VIII. 
CONTINUOUS  GIRDERS. 

§  244.  Fundamental  Principles.  —  A  continuous  girder  is 
one  that  is  continuous  over  one  or  more  supports  ;  i.e.,  one  that 
has  at  least  one  support  in  addition  to  those  at  the  ends.  The 
principle  of  continuity  is,  that  the  neutral  line  is  throughout  a 
continuous  curve  over  the  supports,  the  tangent  to  one  branch 
of  the  curve  at  the  support  being  a  prolongation  of  the  tangent 
to  the  other  branch. 

Whereas,  in  the  girder  supported  at  the  ends,  the  bending- 
moment  at  the  support  is  zero,  in  the  continuous  girder  there 
is  a  bending-moment  at  the  support,  where  the  girder  is  con- 
tinuous. There  is  also  a  shearing-force  at  each  side  of  the 
support,  the  sum  of  the  shearing-forces  on  the  two  sides  of 
any  one  support  forming  the  supporting-force. 

In  this  chapter  will  be  given  the  general  methods  of  deter- 
mining the  bending-moments,  slopes,  and  deflections  of  con- 
tinuous girders. 

i°.  When  the  loads  are  distributed. 

2°.  When  the  loads  are  all  concentrated. 

3°.  When  there  are  both  distributed  and  concentrated  loads. 

It  is  believed  that  the  reader  will  thus  have  the  means  of 
solving  all  cases  of  continuous  girders,  and  that,  whenever  it 
is  desirable  to  have  a  set  of  simplified  formulae  for  a  small  but 


744  APPLIED   MECHANICS. 

definite  number  of  spans,  or  for  some  special  proportions  or 
distribution  of  the  load,  he  will  be  able  to  deduce  such  simpli- 
fied formulae  from  the  more  general  ones. 

§  245.  Distributed  Loads.  —  In  this  case  we  assume  that 
all  the  loads  are  distributed,  whether  they  are  uniformly  dis- 
tributed or  not.  The  first  step  to  be  taken  is,  to  find  the  bend- 
ing-moment  over  each  support :  this  is  done  by  using  what  is 
known  as  the  "three-moment  equation"  which  we  shall  now 
proceed  to  deduce  ;  and,  in  the  course  of  the  reasoning  by  which 
we  deduce  it,  we  shall  derive  a  number  of  useful  equations,  ex- 
pressing bending-moment,  shearing-force,  slope,  deflection,  etc., 
at  various  points. 

E  B  o  x  c  A 


PIG.  247. 

For  the  purpose  in  view,  let  us  assume  our  origin  at  O 
(Fig.  247),  and  let 

MI  =  bending-moment  at  B. 

M2  =  bending-moment  at  O. 

M3  —  bending-moment  at  A. 

A      =  OA. 

/_,    =  OB. 

F0   •=  shearing-force  just  to  the  right  of  O. 

F_0  =  shearing-force  just  to  the  left  of  O. 

Ft    =  shearing-force  at  distance  x  to  the  right  of  origin. 

F_I  =  shearing-force  at  distance  x  to  the  left  of  origin. 

Shear  is  taken  as  positive  when  the  tendency  is  to  slide  tha 
part  remote  from  the  origin  downwards. 

If  £„  =  supporting-force  at  O, 

So  =  F0  +  /I.. 


DISTRIBUTED  LOADS.  745 

Beginning,  now,  by  taking  O  as  origin,  and  x  positive  to  the 
right,  — 

Let  OC  —  x. 

CD  =  v  =.  deflection  at  distance  x  from  origin. 
w     =  load  per  unit  of  length  (either  constant,  or  vari- 
able with  x). 

We  shall  then  have,  from  the  principles  of  the  common 
theory  of  beams, 


—  I      wdx;  (i) 

«/0 


i.e.,  the  shearing-force  at  a  distance  x  to  the  right  of  O  is 
found  by  subtracting  from  the  shearing-force  just  to  the  right 
of  O  the  sum  of  the  loads  between  the  section  at  x  and  the 
support ;  and  this  sum  is 


X 

wdx. 


In  a  similar  manner,  if  we  were  to  take  origin  at  O,  and 

positive  to  the  left,  we  should  have 


'_0  -  f  *  wdx.  (a) 

t/o 


In  §  204  we  found  the  equation 

dM       ., 


dM     ,,     r*    . 

•     —r-  =  A>  — •  I      wax. 
doc  J0 


Hence,  integrating  between  x  —  o  and  x  =  ;r,  and  observing, 
that,  when  *  =  o,  M  =  J/2,  we  have 


M  -  M2  = 


-  I      ] 

t/o     t/o 


746  APPLIED   MECHANICS. 

which  reduces  to 

M=  M2  +  Fox-  f*C*wdx*i  (3) 

Jo      Jo 

or,  in  words,  — 

The  bending-moment  at  a  distance  x  to  the  right  of  O  is 
equal  to  the  bending-moment  over  the  support  at  the  origin, 
plus  the  product  of  the  shearing-force  just  to  the  right  of  the 
origin  by  the  distance  of  the  section  from  the  origin,  minus 
the  sum  of  the  moments  of  the  loads  between  the  section  and 
the  support  about  the  section. 

Observe  that  this  sum  of  the  moments  of  the  loads  between 
the  section  and  the  support  about  the  section  has,  for  its  math- 
ematical equivalent,  the  expression 


/7 

Jo     Jo 


wdx*; 


and,  as  a  particular  instance,  it  may  be  noted,  that  when  the 
load  is  uniformly  distributed,  and  hence  w  is  constant,  this  will 
reduce  to 

WX2  ,         ^  X 


wx  being  the  load  between  the  section  and  the  support,  and  - 

being  the  leverage  of  its  resultant. 
Now  write,  for  brevity, 


wdx*  =  m; 

Jo     Jo 

then 


/T 

«/0       «/0 


M  =  M2  4-  Fjc  —  m.  (4) 

Now,  from  §  194,  we  have 

d*v  =  M 
dx>  =!  Ef 


DISTRIBUTED    LOADS.  747 


Let  a,     =  slope  at  distance  x  to  the  right  of  the  origin. 
a..,  =  slope  at  distance  x  to  the  left  of  the  origin. 
Oo     =  value  of  a,  when  x  —  o. 
a_0  =  value  of  a_ ,  when  JT  =  o. 
Then 


where  c  is  an  arbitrary  constant,  to  be  determined  from   the 
conditions  of  the  problem. 

If,  now,  we  substitute  for  M  its  value  M2  -\-  F0x  --  m,  we 
shall  have 

tana,  =  ^  =  J 


To  determine  c,  observe,  that,  when  x  =  o,  at 

/.     c  =  tana0 


tanaj  =  —  ==  tan 
dx 


/dx 
—, 
.    &J- 


Integrate  again,  and  observe,  that,  when  x  —  o,  v  —  o,  and  we 
obtain 


•/Y 

t/O       t/O 


74$  APPLIED   MECHANICS 


Now  write,  for  the  sake  of  brevity, 

m  =    I      I     wdx2s     m,  =    I    '    /    wdx2,  >»_,  =     I  I     wdx*, 

t/o     Jo  Jo      t/o  «/0  t/o 

r*  r*  '  dx2  rl*  rx  dx*  r'-1  r*dx* 

n  =  J.  I  zi'  "'  -  J-  A  ^  "-  =  J.  ^  ^' 


rr  rJ/«^    „  .     /•'•  fr;«^     „         f'--  /"^^ 
"J0  Jo   ^'       "Jo  Jo  !/*•      '  =  1     J«  ~E 


the  last  four  being  derived    by  taking  x  positive  to  the  left. 
We  shall  have 

v  =  ^tana0  +  M2n  4-  F0q  —  V;  (7) 


and,  if  vl  =  deflection  at  A  =  vertical  height  of  A  above  O,  we 
shall  have,  by  substituting  /t  for  x  in  (7), 

vl  —  /,  tan  a0  +  M2nl  4-  F0ql  —  V^. 

Now,  if  we  assume  any  horizontal  datum  line  entirely  below  all 
the  points  of  support,  and  let  the  height  of  B  above  this  line  be 
j/3,  that  of  At  ya,  and  thaf  of  (9,  y0,  etc.,  we  shall  have 


ya  -  y0  =  /  .an  «0  4-  M2n,  4-  F0q,  -  Vv  (8) 

And,  if  we  put  x  =  /x  in  (4),  we  shall  have 
J/3  =  M2  4-  ^oA  -  ^i 


(9) 


and,  if  we  substitute  this  value  of  F0  in  (8),  we  obtain,  by  redu- 

cing, 

y.  -  y.  =  /,  ,an  „.  + 


DISTRIBUTED   LOADS.  749 

p.nd,  solving  for  tan  Oo,  we  obtain 

ya  -  }'Q   , 
tan«0  =     --    + 


This  expression  gives  us  the  tangent  of  the  slope  at  O  in  span 
OA  ;  and  equation  (9)  gives  us  the  shearing-force  just  to  the  right 
of  O  in  span  OA,  in  terms  of  M2,  M3,  and  known  quantities. 

If  we  were  to  take  the  origin  at  O,  as  before,  and  x  positive 
to  the  left  instead  of  the  right,  we  should  have,  in  place  of  (4), 

M  —  M2  -f  F_0x  —  m;  (n) 

in  place  of  (9), 

M,  -  M2  +  **_, 
*_.--        -jr-       -5  (12) 

and  in  place  of  (10), 

tana_0  =  — ?  -f  j 

*  — i 


But,  since  the  girder  is  continuous,  we  must  have  the  tangent  at 
O  to  the  left-hand  part,  a  prolongation  of  the  tangent  at  O  to 
the  right-hand  part,  as  shown  in  Fig.  248. 
Hence  we  must  have 


/.     tana_0  -f-  tana0  =  o. 

Hence,  adding  (io)-and  (13),  we 
have 


r    T  r  i 


and  this  is  the  "three-moment  equation"  for  the  case  of  a  dis- 
tributed load,  whether  it  be  uniformly  distributed  or  otherwise. 


75°  A2TL2ED   MECHANICS. 


CASE    WHEN    SUPPORTS    ARE    ON  THE    SAME    LEVEL. 

When  the  supports  are  all  on  the  same  level,  then  yA 
=  y0,  and  the  three-moment  equation  becomes 


" 


MANNER  OF   USING   THE   THREE-MOMENT   EQUATION. 

When  the  dimensions  and  load  of  the  girder  are  known,  all 
the  quantities  in  the  three-moment  equation,  whether  we  use 
(14)  or  (15),  are  known,  except  the  three  bending-moments,  Mlt 
M2,  and  My 

Suppose,  now,  the  girder  to  have  any  number  of  (say,  seven) 
points  of  support ;  then,  by  taking  the  origin  at  B  (Fig.  247), 
we  obtain  one  equation  between  the  bending-moments  at  £,  B, 
and  O,  the  first  of  which,  if  E  is  an  end  support,  is  zero.  Next 
take  the  origin  at  O,  and  we  obtain  one  equation  between  the 
three  bending-moments  at  B,  O,  and  A;  and  so,  continuing,  we 
obtain  five  equations  between  five  unknown  quantities. 

Solving  these,  we  obtain  the  bending-moments  over  the 
'supports ;  and  from  these  bending-moments,  after  they  are 
found,  we  can  obtain  the  shearing-forces,  bending-moments, 
slopes,  and  deflections,  by  using  the  equations  deduced  in  the 
course  of  the  reasoning  for  the  three-moment  equation,  as  equa- 
tions (4),  (5),  (7),  (9),  and  (10). 

SPECIAL  CASE, 

when,  the  supports  being  all  on  the  same  level,  the  load  on  any 
one  span  is  uniformly  distributed  over  that  span,  and  when  the 
girder  is  of  uniform  section  throughout. 


DISTRIBUTED   LOADS. 


Let  wt  —  load  per  unit  of  length  on  span  OA,  origin  at  O. 
w_t  =  load  per  unit  of  length  on  span  OB,  origin  at  O. 
I  —  the  constant  moment  of  inertia  of  the  section 

Then 

wJS  w-j.f 


2Ef 


-  6Ef 

V    _    ?^i/i4 

1    ~~  T~>  7* 

2^EI 

With  these  substitutions,  the  three-moment  equation,  either 
(14)  or  (15),  becomes 

This  is  a  simpler  form  of  the  three-moment  equation,  applicable 
to  this  particular  case  only. 

EXAMPLE    I.  —  Suppose   we   have   a   continuous   girder  of 
uniform  section,  uni- 
formly   loaded,    and    * ~* * ;  * 

of  three  equal  spans, 

to  find  MB  and  Mc,  also  the  supporting-forces,  shearing-forces, 

bencling-moments,  slopes,  and  deflections  throughout. 

Solution.  —  Take  the  origin  at  B,  and  we  have 

since 


equation  (16)  gives 

o  .-.    M*  =  Afc  =  -— . 

10 


752  APPLIED   MECHANICS. 


Next,  to  find  the  shearing-forces,  we  have,  from  (9), 


10  10  2  Wl 


equals  shearing-force  just  to  the  right  of  B  or  left  of  C. 
Shearing-force  just  to  the  right  of  C  or  left  of  B  = 

ze//2       w/2 
10          2  . 


Hence  supporting-forces  are 

SK  =  Sc=  (     -f-  \}wl= 


Bending-moment   in   span  AB  at  distance  x  from  A,  cr  in 
span  £7?  at  distance  x  from  Z?, 


Bending-moment    in    middle    span    at  a  distance  ^r  from  ^  or 
from  6T, 

7£//2  Wlx-         WO? 

J/—    -----  (____    ----  , 

IO  2  2 

Shearing-force  in  span  AB  or  CD  at  a  distance  -r  from  /i 
or  Z>, 

^  =  |-7£//  —  wjf. 

Shearing-force  in  middle  span  at  distance  x  from  B  or  (7, 

7P  ^ 

r  =         —  wx. 

2 

Maximum  bending-moment  in  span  BC  (when  x  —  -), 

7£//2  Wl2  Wl2  Wl* 

/i/y      —  — 

J-'J-G     -  „          -  • 

10          4  8          40 


DISTRIBUTED  LOADS.  753 

Maximum  bending-moment  in  span  AB  or  CD, 
*  =  ¥, 


25  5°  25 

Hence   the   greatest   bending-moment  to  which  the  girder  is 

wl2 

subjected  is  that  at  B  or  C,  and  its  amount  is  —  . 

10 

Slope  at  B  in  middle  span,  from  equation  (10), 


io\6JSl 


i 

60 


i          i  \  _ 
0       24/ 


which  denotes  an  upward  slope  at  B  towards  the  right.  In  the 
sime  way,  the  girder  slopes  upwards  at  C  towards  the  left. 
The  slopes  at  B  and  C  in  the  end  spans  are,  of  course,  down- 
wards. 

Slope  in  the  middle  span  at  a  distance  x  from  j?, 

dv         T   (      wl2      ,    wlx2       wx^> ) 

tan  a  =  —  =  —  < x  -\ >  4-  c. 

dx       EI\       10  4  61 

When  x  =,  o, 

wfi                                  wfi 
tana  =  -j- /.    c  =  — r. 

\2OEI  \2QEI 

w  (   /3         I2x       lx2       x*\ 

.*.    tan  a  =  —  I 1 v 

EI[  1 20        10         4         6  j 


\2QEI 


Deflection  =  v  =  -    —(l*x  —  6/2.r*   f 


754  APPLIED   MECHANICS. 

In  order  to  make  plain  all  methods  of  proceeding,  the  slope 
in  the  end  spans  will  be  found  in  two  different  ways,  as 
follows  :  — 

For  bending-moment,  slope,  and  deflection  in  left-hand  span 
at  a  distance  x  from  7?  (or  in  the  right-hand  span  at  distance  x 
from  C)9  we  have 

wl2       3  w& 

M  —  ----  h  -wlx  —  •  —  -. 
10        5  2 


_  dv  _     i   (      wl2x       ^wlx*       wx*  \ 
*~'  ~w"'     ~^~       Tf" 


When  x  =  o, 


tan  a  =  --  —  -  /.    c  =  — 


120^7  I20.E7 


/&        w  (       /3         /2*    ,     3  .          x*) 

.-.    tan  a  =  —  =  —I -  +  —  I**  -  -7  \ 

dx       EJ{      120        10        10  6  ) 


w 


12OEI 


Deflection  =  v  =  -  ^-—  (-/^  -  6/2Jc2  -f  i2lx*  - 

120^7 


We  may,  on  the  other  hand,  accomplish  the  same  object  by 
finding  the  slope  and  deflection  in  left-hand  span  at  distance 
x  from  A,  or  in  right-hand  span  at  distance  x  from  D,  as 
follows  :  — 

dv         i     C  ( 2     ,        wx2 }   ,          w  ( L\2       x*  \ 

tan  a  =  —  =  —  I   \  -wlx \dx  =  —\ — V  -}-  c. 

dx       EIJ    ($  2    )  EI(  5          6  ) 

When  x  =  /, 

tana  =     wl* 


wl*  wl*  wl* 

•  ,___  _  — I  y«  •  f     •••       i-— ^     

120EI~    30^7  40^7 


DISTRIBUTED   LOADS.  755 


.'.     tan  a 


-  &L  =.  yL\  —  ii  4.  ^!  _  £!l 
"  dx  ~  EI\      40        5      "  6  ) 


\2QtEI 


The  figure  shows  the  mode  of  bending  of  the  girder. 


FIG.  250. 

To  find  the  greatest  deflection  in  either  span,  put  the  ex- 
pression for  the  slope  equal  to  zero,  and  find  x  by  the  ordinary 
methods  for  solving  an  equation  of  the  third  degree,  and  then 
substitute  this  value  in  the  expression  for  the  deflection. 

EXAMPLE  II.  —  Continuous  girder  of  two  equal  spans,  sec- 
tion uniform,  and  load  uniformly  dis- 
tributed. A  *  * 

~    ,    ^  .  ^    ,  .     .         ,_    „  FIG.  251. 

solution.  —  1  ake  origin  at  B. 
M,  =  J/3  =  MA  =  Mc  =  o,    M2  =  MB,    /,  =  /2  =  /,    w,  =  w2  =  w; 

therefore,  from  equation  (16), 

4/J/B/  +  Jw/3  =  o          .-.     Jlf9  =  -  —  . 

8 

Shearing-force  either  side  of  B  — 

w/2       a//2 


Supporting-force  at  B  =  |w/. 
Supporting-force  at  ^4  and  (7  = 
Shear  at  distance  x  from  A  or 


750  APPLIED   MECHANICS. 

Bending-moment  at  distance  x  from  A  or  C, 

-i  wx2 

M  =  -wlx  —  -- 

8  2 

Maximum  bending-moment  occurs  when  x  =  f/, 


64  128  128 

Hence  greatest  bending-moment  to  which  the  girder  is  sub« 

wl2 

jected  is  that  at  B,  and  its  magnitude  is  -  . 

8 

Slope  at  B,  from  equation  (6), 


wl2  (         I    \         wl* 

tanaB  =  tana_B  = I  _  1  —  

8    \      $EIi        1 2E I 


24^7 


as  was  to  be  expected. 

Slope  at  distance  x  from  A  in  span  AB, 


When  x  =  /,      a  =  o  ; 


C  =  -  wlZ 


48^7 
~EI  (16*      "  ~6  ~~  48 


Deflection, 


For  maximum  deflection,  we  have 


----  =  o 
16  6       48 

.*.     x  —  o.44/. 


DISTRIBUTED   LOADS.  757 

Maximum  deflection  =  —  —  \  —  i  4-  3  (0.44)  2  —  2(0.44)31  (0.44) 
48.fi/  (  ) 

w/4 

=  -0.0054— 

EXAMPLE  III.  —  In  order  to  solve  a  case  where  no  simplifi 
cations  enter,  on  account  of  symmetry  or  otherwise,  we  will 
take  a  continuous  girder  of  five  spans  (as  shown  in  the  figure), 
the  spans  varying  in  length  from  3/  to  jl  ;  the  loads  being 
uniformly  distributed,  and  varying  in  intensity  from  3^  on  the 
longest  span  to  721;  on  the  shortest  ;  the  beam  being  of  uniform 
section. 

A         :i/         B  41  C  _  51  _  D  _  6J  _  E  _  Tl  _  F 

A         1w        A  6w  A  bv>  A  4w  A  3w  A 

FIG.  252. 

For  this  case  we  can  use  equation  (16). 
Origin  at  B, 

o  +  i4/J/B  +  4/J/c  +  £[7^(27/3)  +  607(64/3)]  =  o, 
or 

56J/B  -f  i6J/c  =  —  5730^. 
Origin  at  C, 

4/ATB  +  iS/Mc  +  5/J/D  +  i«//3[6(64)  -f- 
or 

I6J/B  -|-    72J/C  -j- 

Origin  at  A 


or 

c  4-  88J/D  -f 


Origin  at  E, 


E  -f  —[4(216)  +  3(343)]  =  o, 
4 


or 

-f- 


758  APPLIED   MECHANICS. 

The  four  equations  are  : 

$6MK  -h  i6Jlfc  =  -  573^/2.  (i) 

I6J/B   +    72J/C   4-    20J/D  =  —  I009?£'/2.  (2) 

4-  88J/D  +     24^  =  —  1  4890/7*  .  (3) 

D  4-  I04J/E  =  —  i893w/2.  (4) 


Eliminate  J/E  between  (3)  and  (4),  and  we  obtain 

i3oJ/c  4-  536^D  =  -6839W/2.  (5) 

Eliminate  Mv  between  (2)  and  (5),  and  we  obtain 

2144^/3  4-  8998^  =  —  roioiio//2.  (6) 

Eliminate  J/c  between  (i)  and  (6),  and  we  obtain 
234792^/13=   — 


.'.  from  (i),  J/c  =  --  9.4299^/2, 
from  (5),  J/D  =  —  10.4  7  22«>/2, 
from  (4),  J/E  =  -i 

Shearing-force  just  to  the  right  of 
-o  -  7.5379  +  3i.5 


4 

-10.4722    4-   9.4299    4-   62.5 
C  —  -  —wl—  12.29152^4 

^  =  -  -5.7853  +-o.47^  +.  7«w  _  „ 


'5.7853 


DISTRIBUTED   LOADS. 


759 


Shearing-force  to  the  left  of 
7-5379  +  3i-5 


13.01260/7, 


„        -7-5379  +  94299  +  48 

L  =  -  —  wl      =  12.47300/7, 

-9.4299  +  10.4722  +  62.5 
D—  -  -  —  o'7  =  1  2.  70840/4 

-10.4722  +  15-7853  +  72   . 

E  =  -  6          -  o/7  =  12.88550/7, 

,,,        ~15-7853  4-   735     , 
-r  =  —  —        —a//  =    8.24500'/. 

Supporting-force  at 


C  =  24.76450/7,  E  =  25.64050/7, 

«#=24.5396a/7,  /?=:     23.82290/7,  F    =        8.24500/7. 

Shearing-force  at  distance  x  to  the  right  of 

A  in  section  AB  =  7.98740/7  —  70/0:, 
B  in  section  BC  =•  11.52700/7  —  6wx, 
C  in  section  CZ>  =  12.29150/7  —  5760:, 
D  in  section  Z>,£  =  11.11450/7  —  40/3:, 
^  in  section  EF  =  12.75500/7  —  yjux. 

Bending-moment  at  distance  x  from 

A  in  section  ^  =  +     7.98740/7*  - 

B  in  section  BC  —  —    7.53790/72  +  11.52700/7^  - 


Cin  section  CD  =  —  9.42990/72  +  12.29150/70.-  -  — , 
Z>in  section  Z>j£  =  -10.47220/72  +  11.11450/7*  -  ^wx\ 
E  in  section  EF  =  -15. 78530^  +  12.75500/7*  - 


760  APPLIED   MECHANICS. 


For  the  sections  of  maximum  bending-moments  (put  shear- 
ing-force =  o),  — 

In  AB,  x  =  T.I4IO/,* 
In  BC,  x  =  1.92111  f 
In  CD,  x  ==  2.4583// 
In  DE,  x  =  2.77S6// 
In  EF,  x  =  4.25177. 

Hence   the   maximum    bending-moments  are  respectively, 

in  — 

Section  AB, 


+4.5570^. 

Section  BC, 

—  7-5379^ 
Section  CD, 

—  9.4299101*  4-  ^^(12.  29157  —  %x)  =    5. 
Section 


Section  EF, 


Values  of  tan  Oo  =  slope  in  every  case  in  the  span,  towards 

/  q,        n\        Mtfi        msft        V^ 
a0  =  M2(j-2  -jj-    —-—  +  -. 


the  right. 


tan 
Slope  at  B, 


tan  a,  =  -  7-5379 


0.337'^- 


DISTKJBUTED    LOADS. 


Slope  at  C, 

W^f3 
tanac    .      -9-4^99 


625 
~ 


Slope  at  Z>, 

O//3  Z£//3 

tanaD  =  -10.4722-^(1  -  3)  +  15-7853^7 


Slope  at  E, 

w^ll       7\       1029  w/3      343  w/3 

-  '5-7853^^--;- 


El 


tan  aE   =  — 


12 


The  manner  of  bending,  very  much  exaggerated,  is   shown   in 
the  accompanying  figure. 


FIG.  253. 

Slope  at  A  =  -4.096—,     slope  at  F  =  +23.4578— J5 

El  £. 

For  the  deduction,  see  what  follows. 
Slopes  in  General. 

Span  AB,  origin  at  A, 


tana  =  -=?  ^3.9937^  -  -; 


When  ^r  =  3/,  tan  a  =  0.3371 


El  ' 


wl* 


•'•     ^y(35-9433  -  3i*5)  +  ^  =  o. 


.*.     c  =  —4.106- 


/.     tana  =  ^7[3-9937^2  ~  \&  -  4-10^*}- 


762  APPLIED   MECHANICS. 

Span  BC,  origin  at  B, 

wfi  wl2x  ,     wlx2 

tana  =  0.337!—  -  7.5379-^r  +  5-7635^  •-  -j 

Span  CD,  origin  at  C, 

tana  —  ~^j\  —1-5983/3  —  9-4299/2*  4-  6.i458/r2  —  - 
Span  DE,  origin  at  D, 

W     (  2       ) 

tana  =  )  0.7297/3  —  io.4722/2^  -\-  5-55725/r2 ^sv. 

El (  3     ) 

Span  EF,  origin  at  E, 

tana  =  — )  —6.0426/3  —  i5.7853/2^:  4-  6.3775/r2  —   — >. 

When  x  —  ;/, 

tana  =  ^—(  —  6.0426  —  110.4971  4-  312.4975  —  172.5) 

=    T23.4c,S^3 

Deflections. 
Span  ^4/?, 

w  (                        7  ) 

v  =  ~^7'{  1.3312/^3 **  —  4.io6/3^>» 

y^y  (  24  j 

Span  y?(T, 

z,  =  ^-{0.3371/3*  —  3.7689/2^2  4-  1.9211^3  _  fElL 
yj/  (  4  ) 

Span  £Z?, 

a/  (  5 

z;  =  ^7-!  —1.5983/3*  —  4-7I49/2*2  4-  2.0486/^3 

y^y  (  24 

Span  DE, 


»=  ^-  \  0.7297/3^  -5.2361/^4-  1.8524  +  ^3  -£ 

y^y  o 


Span  EF, 


_^_  I  -6.0426/3*  -  7.8927/z**  4-  2.1258/^3  -  ^ 
y5"/  1  8 


CONTINUOUS   GIRDER  WITH  CONCENTRATED  LOADS.    763 


The  maximum  deflections  can  be  obtained  by  putting  the 
slopes  equal  to  zero,  as  before. 

§  246.  Continuous  Girder  with  Concentrated  Loads.  — 
For  our  next  general  case,  we  will- 

take  that  where  there  are  no  dis-     — ^ : 

tributed  loads,  but  where  all  the     n-1 

loads   are   concentrated   at   single 

points,   and   the    section    uniform  Wn~FiG  2      W" 

throughout ;  and  we  will  begin  by 

assuming  only  one  concentrated  load  on  each  span. 

Let  the  support  marked  n  —  i  be  the  (n  —  i)tb  support,  and 
the  length  of  the  (n  —  i)th  span  be  4_x;  let  the  load  on  this 
span  be  Wn  _  „  and  likewise  for  the  other  spans.  Assume  the 
origin  at  «,  and  let 

Fn      =  shearing-force  just  to  the  right  of  n. 
F_n  —  shearing-force  just  to  the  left  of  n. 
Fl      =  shearing-force  at  distance  x  to  the  right  of  n. 
,F_  j  =  shearing-force  at  distance  x  to  the  left  of  n. 

Shear  is  taken  as  positive  when  the  tendency  is  to  slide  the 
part  remote  from  the  origin  downwards. 

If  Sn  =  supporting-force  at  «, 


Sn  =  Fn  +  F.n.  (,) 

Let,  also,  xn  =  distance  from  origin  to  point  of  application 
of  load  WMt  and  let  xn_^  =  distance  from  origin  to  point  of 
application  of  load  Wn_l. 

Take  x  positive  to  the  right.     Then,  for 

x  <  xn,    ft  =  Fn ; 

Moreover,  we  have 

dM 


764  APPLIED   MECHANICS. 

hence,  by  integration,  for 

/W  Px 

x<xn,    I    dM  =    I    Fndx; 
JM,.  Jo 

PM  Px  Px 

x>xn,        dM  =    \    Fn<tx-    I     Wndx  +  c; 

UMn  Jo  Jo 

the  value  of  c  being  determined  from  the  condition,  that,  when 
x  =  xn  the  two  results  must  be  identical.     Hence  we  have,  for 


,      M  =  Mn  +  Fnx; 

X>Xnt        M  =   Mn   +  FnX   -    Wn(x   -  Xn). 

Make  x  =  4  in  the  last  equation,  and  we  have 

M*  +  l  =  M»  +  Fnln  -  Wn(ln  -  xn).  (4) 

Now  let  4  —  *n  =  an>  and  (4)  becomes 


hence 


p  —  Mn  +  '  ""  Mn          n*  (6) 

*» 


Moreover,  we  have,  as  before, 
d*v       M 


/being  a  constant. 
Let,  as  before,  — 

aj    =  slope  at  distance  x  to  the  right  of  origin. 
a_,  =  slope  at  distance  x  to  the  left  of  origin. 
an    ==  value  of  a,  when  x  =  o. 
a_^  =  value  of  a_j  when  x  =  o. 


CONTINUOUS  GIRDER  WITH  CONCENTRATED   LOADS.   765 

Then  by  integration,  determining  the  constant  in  the  same 

way  as  in  (3),  we  have,  for 


x  <  xn,     EJ(tenat  -  tana*)  =  M^x  +  F 


x  >  xn,    ^/(tana,  -  tana*)  =  Mnx  +  Fn—  -  ^-^- ^ 

22 


(7) 
^/(tana,  -  tana.)  =  Mnx  +  Fn-  -  "*v"  ""  *"'. 

2  2 

Hence 

x  <xn,    El —  =  El  tan  &„  +  Mnx  -f  Fn—  \ 
ax  2 


j-,T  r,T^  ,, 

x  >  xn,     El—  =  ^7  tan  a*  +  Mnx 


Integrate  again,  and  determine  constants  in  the  same  way, 
and  for 


x2 
x  <  xn,      EIv  =  EIx  tan  o»  -f-  Mn — 

2 

x  >  xn,      EIv  =  EIx  tan  an  -{-  Mn— 

2 


(8) 


Make  x  =  ln  in  the  last  equation,  and  denote  the  heights 
of  the  supports  above  the  datum  line  in  the  same  way  as  in 

§  245,  and  we  have 

EI(yn  + ,  —  yn)  =  EIln  tan  an 

42  43        Wn(ln  —  #«)3 

'         ^  x  /*  *      WX 

20  6 

Substitute  for  /„  —  ^«,  ^«,  and  for  Fn,  its  value  from  (6),  and  we 

have 

EI(yn  +  i  —  yn)  =  EIln  tan  a^ 

+  Mn—  +  Mn  +  l^  H J-^(42  —  #«*)•  (10) 

3  6-6 

Hence 


-f 
6 


l  ~n.     (n) 
4 


766  APPLIED  MECHANICS. 

Now,  if  we  take  origin  at  n  and  x  positive  to  the  left,  we 
should  obtain,  instead  of  (n), 


s=fi=.'<4->-  «.-v>-^)f^}  («.) 

Now  add  (n)  and  (12),  and  observe,  that,  since  the  girder  is 
continuous, 

tan  a^  -f  tan  a_«  =  o, 

and  we  obtain 

*(4  + 4-0 +*-,%•• +  *f* 


64  _  i 


and  this  is  the  "three-moment  equation"  for  the  case  of  a 
single  concentrated  load  on  each  span,  and  a  uniform  section. 
When  the  supports  are  all  on  the  same  level,  this  becomes 


(4  +  4.,)  +  ^-r  +  +  -(42  -  a*) 

3  6  6  64 

+  -^(4_.°  -«„_.')  =o.  (.4) 


Either  of  these  equations  can  be  used  (when  it  is  appli- 
cable) just  as  the  three-moment  equation  was  used  in  the  case 
of  distributed  loads. 


CASE   OF  MORE    THAN  ONE  LOAD   ON  EACH  SPAN. 


CASE  OF   MORE  THAN   ONE  LOAD  ON   EACH  SPAN. 

When  there  is  more  than  one  load  on  each  span,  the  three- 
moment  equation  becomes  as  follows  :  — 


^2(4  +  4-0  +  ^->^'  +  *•+£ 

3  66 

-  a«3)  +  ^Wn-*a"-l(ln-?  -  an. 
64 -i 

yn  +  lTyn  +?»-;->*[.  (I5) 


In  using  these  equations  for  concentrated  loads,  we  can 
determine  the  moments  over  the  supports ;  but  we  must  observe, 
that,  in  getting  slopes  and  deflections,  bending-moments,  etc., 
the  algebraic  expressions  that  represent  them  are  different  on 
the  two  sides  of  any  one  load,  and  hence  we  must  deduce  new 
values  whenever  we  pass  a  load,  determining  the  constants  for 
our  integration  to  correspond. 

EXAMPLE.  —  Given  a  continuous  girder  of  three  spans,  the 
middle  span  =  20  feet,  each  end  span  =  15  feet;  supports  on 
same  level.  The  only  loads  on  the  girder  are  two ;  viz.,  a  load 
of  5000  Ibs.  at  5  feet  from  the  left-hand  end,  and  one  of  4000 
Ibs.  5  feet  from  the  right-hand  end.  The  supports  are  lettered 
from  left  to  right,  A,  B,  C,  D,  respectively.  Find  the  greatest 
bending-moment  and  greatest  deflection. 

Solution.  —  Origin  at  By 

JLf  Jif  cnoo   v    c 

_^o  +  ,S)+f<(2o)+5^(MS-,s)-»    (,) 
Origin  at  C, 

—(20  -h  15)  +  -^(ao)  -f  45?? -(**$  —  15)  «•«.     (t) 

*"  £.       ^  '  £.      ».>       -  -*        *  *^  ii S  ^      F 


;68 


APPLIED  MECHANICS. 


These  reduce  to 


4- 
2oMB  4- 


1000000 


,   800000 
H ss  o 


Jf,  =3  —4000  foot-lbs. 

Afc  SB  —2667  foot-lbs. 


Shearing-forces. 

3067,  /LC  =  -67. 


Supporting-forces. 

&  =  3067. 


Slopes  at  supports. 

tana.    --52 


35556 
El  ' 


»SB  =  2000.         tan  aB    =  4- 
I*t    ss      67*    -^l-D ==  2489.         »Sc  ==  *444«          tan  oc     =  — 

JLI 

S  —      8  tan  48889 

Span  -<47?,  origin  at  A, 

*  <  5,    ^f  =  3067*. 

*>  5,    M=  3067*  -  5ooo(*  —  5)  =  25000  —  1933*. 

Maximum  bending-moment  occurs  when  x  =  5  and  therefore 

^o  =   15333- 

*  <  5,    El  tan  o,  =  -59444  4-  I533*2  > 

*  >  5,    El  tan  a,  =  25000*  —  967**  4-  c. 

Determine  c  by  condition,  that,  when  x  =  5,  these  two  become 
equal; 


•"•    x>  5>    -57 tan  a,  =  —121944  4-  25000,1;  —  9674*. 
For  deflections, 

x  <  5,     .£70  =  -  59444*  4-  511*3 ; 

x  >  5,    7sY#  =  —121944.2:  4-  12500**  —  322^  4-  <"• 
Determine  ^  from  condition,  that,  when  x  =  15,  v  =  o; 
.*.     <•  sss  104167 ; 

•"•    *  >  5»    ^-^  ==  104167  —  121944*  4-  12500**  —  322*'. 


CASE   OF  MORE    THAN  ONE  LOAD   ON  EACH  SPAN. 


For  maximum  deflection,  equate  slope  to  zero,  and  find  x. 
We  find  it  at  x  =  6.53. 
/.    EIvQ  =  —249531. 

Span  BC,  origin  at  B, 

M  =  —4000  -f-  67*, 
El  tan  a,  ==  35557    —  4000*  -f  33**, 
EIv          =  35557*  —  2000*2  -f-  n#». 

For  maximum  deflection,  equate  slope  to  zero,  and  find  J» 

,     We  find  it  at  x  =  9.78. 

.*.    Jzlvc,  —  i  (.6  740. 
Span  CD,  origin  at  C, 
x<io,M—  —2667  -f  1511.*  ,• 

x>  10,  M  =  —2667  +  1511*  —  4ooo(^  —  *o)  =»  37333  —  2489*; 
x  <  10,  El  tan  ar  =  —31111  —  2667.3:  +  756^  ; 
x>  10,  .S/tan  ax  =  —231111  -f  37333*  —  1245**. 

For  deflections, 

x<  10,  EIv  =  —  3IIHJC  —     I334-*2  +  252^5 

x>  10,  EIv  =  —231111.*:  +  18667^  —  415^  -f-  f. 

When  x  —  15,  v  =  o; 

/.    #>  10,    .S/z/  =  -37132785  —  231111^+18667^—  4T5*». 
For  maximum  deflection,  equate  slope  to  zero,  and  find  x. 

We  find  it  at  x  =  8.41. 
.*.     EIvQ  =  —24506. 

Hence  greatest  bending-moment  and  greatest  deflection  are 
both  in  span  AB. 

Observe,  that,  since  we  have  used  one  foot  as  our  unit  of 
measure,  all  dimensions  must  be  taken  in  feet,  and  the  value 
of  E  is  also  144  times  that  ordinarily  given. 


77O  APPLIED  MECHANICS. 

§  247.  Continuous  Girder,  with  both  Distributed  and 
Concentrated  Loads.  —  In  this  case  we  may  either  calculate 
the  bending-moments,  slopes,  and  deflections  due  to  each  sepa- 
rately, and  then  add  the  results  with  their  proper  signs,  or  we 
may  modify  the  solution  that  was  used  for  the  case  of  a  dis- 
tributed load,  so  as  to  extend  its  applicability  to  this  case. 

Let  W  represent  any  one  concentrated  load,  and  let  x^  rep- 
resent the  distance  of  its  point  of  application  from  the  origin. 
Then,  in  the  general  formulae  deduced  for  the  distributed  load,, 
make  the  following  changes  ;  viz.,  — 

1°.  Instead  of 

m  —  I      I    wdx*, 

«/o    t/o 

put 

m 


=  CX 

t/o    «/o 


since,  as  was  shown,  m  represents  the  sum  of  the  moments  of 
the  loads,  between  the  section  and  the  support,  about  the  sec* 
tion. 

2°.  Instead  of 


Xx  /v  i    rx  f*A          \  fi>& 
1  iJL  JL  •*•}£• 


put 


Jo     Jo      (Jo    Jo  I  El  Jx.  Jxi 


and  make  the  corresponding  changes  in  the  values  of  mlf  m^^ 
Vu  and  F_,,  leaving  n  and  q  just  as  before  ;  then  use  the  same 
three-moment  equation  as  before,  with  these  substitutions,  i.e., 


SPECIAL    CASE. 


SPECIAL    CASE. 

when  the  distributed  load  is  uniformly  distributed  on  each  span, 
but  may  be  different  on  the  different  spans,  and  when  the  girder 
is  of  uniform  section. 

Let  wl    =  weight  per  unit  of  length  on  OA. 
w_i  =  weight  per  unit  of  length  on  OB. 

Denote  by  Wt  any  concentrated  load  on  OA  at  distance  #, 
from  O. 

Denote  by   W_  ,  any  concentrated  load  on  OB  at  distance 
;*•_!  from  O. 

Then  we  shall  have 


v 

and,  as  before, 


Making  these  substitutions  in  the  three-moment  equation, 
and  clearing  fractions,  we  obtain  for  the  case,  when  the  sup- 
ports are  all  on  the  same  level, 


[/x*  -  (4  - 


4'    ^     3' 
4, 


772  APPLIED  MECHANICS. 

CONCENTRATED    AND    DISTRIBUTED    LOADS. 

EXAMPLE.  —  Let  the  girder  be  of  uniform  section,  of  two 
equal  spans,  each  being  10  feet ;  let  the  concentrated  loads  be 
A  B  c  as  shown  in  the 

>]<    4'     >[<    3'  >   figure,    the     dis- 

*  tributed  load  be- 

1000  looo  e 

ing    96    Ibs.  per 

FlG-255-  foot.      Find    the 

value  of  El,  so  that  the  deflection  may  nowhere  exceed  -^  of 
the  span. 

Solution.  —  Use  equation  (12);   and,  in  deducing  value  of 
M&  use  dimensions  in  feet ;  afterwards  use  inches. 
Origin  at  B,  M  =  M  =  o- 

946(iooo  -f-  1000) 
+  3^2000(64)  (6)  -f-  1000(51)  (7)  -h  1000(91)  (3)!  =  o, 

B  -f-  48000  4-  139800  =  o,     or    MB  =  —4695  foot-lbs., 

J/B  =  —56340  inch-lbs.,    MA  =  Mc  =  o. 

mt  =  177600,  m_^  —  201600, 

7200  nt        60  7200 

"*   "'  ''  ~EI'  J^  El'  n~*  =  ~ET' 

288000  qT         20  _  288000 

El  I?       El  El 

175680000  Fx       1464000 


El          /,  El  El 

o  -|-  56340  -f-  177600 
Shear  right  side  of  middle  = -  =  1949.5, 

o  +  56340  -h  201600 
Shear  left  side  of  middle      =  -  -  =  2149.5  * 

Shear  left  end  = 


Shear  right  end  .  ^563^177600 

Middle  supporting-force      a  4099. 


SLOPES.  773 


Bending- Moments  in  Each  Span. 
Span  AB,  origin  at  A, 

810.5*  —  4*2 
or 

810.5*  —  4**  —  2Ooo(#  —  72). 

Span  BCt  origin  at  B, 

—  56340  4-  1949-5*  -  4*% 

—  56340  +  I949-5-*  —  4**  —  iooo(#  —  36), 

—  56340  -+•  I949-5-*  —  4X2  —  iooo(*  —  36)  —  iooo(je  —  84). 

To  ascertain  position  of  the  greatest  bending-moment,  dif- 
ferentiate each  one. 

810.5  —  &#  =  o,  x  =  101.31 ; 

810.5  —  8*  —  2000  =o,  x  =  a  minus  quantity; 

1949.5  —  8*  =  o,  x  =  243.69  ; 

1949.5  —  8*  —  1000  =o,  x  —  118.69; 

1949.5  —  8*  —  1000  —  1000  =  o,  x  —  a  minus  quantity. 

Hence,  in  span  AB,  maximum  bending  is  at  the  load,  and 
its  amount  is 

(810.5)  (72)  -  4(72)  (72)  =  37620. 

Span  BCy  maximum  is  at  right-hand  load,  and  is 
-5634o  +  1949.5 (84)  -  4(84X84)  -  1000(48)  =  31194. 


SLOPES. 

Slope  at  B, 

-56340  (177600) (20)    t    1464000       165600 

El     ^  ~£7~  ~ET 

Slope  and  Deflection  in  Span  AB. 
First  part, 

tana  =  ^ 1 405.25.**  —  -#*>  4-  tana* 
a,,  being. slope  at  A. 


774  APPLIED  MECHANICS. 

Second  part, 

1(4  ) 

tana  =  ^^405.25^  --  x3  —  looar2  -f  144000.3:  >  -f-  1. 

When  x  =  72,  a  is  the  same  in  both  cases  ; 

.-.     JLj  1000(72)*  -  (144000)  (72)!  +  tanoo  -  c  -  o 

£Ll 

5184000 

•'•     '-tana.-    —ft- 
When  x  =  120,  the  second  value  of  tana  becomes  ~w  — 


tan  ao- 


tana,  =  -M-64ii6oo  +  5184000  +  i6S6ooj  =  -Io6p2°°°> 

J2,l  •&•* 

6246000 

~~ET 
Hence  slope  in  first  part  (between  A  and  the  load), 


tana  =  -^r7<  —1062000  +  405 
El  ( 


4     ) 

.25^  --  *3> 
3      ) 


Second  part  (between  B  and  the  load), 
tana  =  -gr^|  —6246000  +  144000*  —  594-7S-*2  —  **  | 

Deflection. 
First  part, 

v  =  -j-j  |  —1062000*  +  135.08^  —  —1. 

Second  part, 

9  =  —  I  —6246000*  H-  72000*1  —   198.25**  —  -  4P4!   -H 


SLOPES.  775 


When  x  =  120,  v  =  o ; 

c  = 

(6246000)  (120)  — 


I2O/  ,0       x  I2 

=  —(1036800)= 

Point  of  greatest  deflection  is  found  by  putting  slope  equal 
zero.  Moreover,  it  is  plain  that  the  greatest  deflection  is  in  the 
first,  and  not  the  second,  part. 

Hence  equation  is 

•f*3  —  405.25^  +  1062000  =  o 
.-.    *=  56".77; 

and,  substituting  this  in  the  expression  for  the  deflection,  we 

obtain 

39037720 


z>0  =  — 


El 


I 20       39037720 

Hence,  putting  -  -  = =rj — ,  we  obtain 

400  jbLl 

El =  130125733. 

If  E  =  1400000,  /  =  92.9 ;  therefore,  if  b  =  3  inches,  h  = 
7  inches. 

Slope  and  Deflection  in  Span  BC. 
Portion  nearest  B, 

i    (                                                          4     ) 
tana  =  -gj\  165600  —  56340.*:  -|-  974.8^ #*>• 

When  x  —  36  inches,  we  obtain 

i                                                                                  661507 
tana  =  -^(165600  —  2028240  +  1263341  —  62208)  = -,    • 

Middle  portion, 

tana  =  2^|  -20340*  +  474-7S*2  ~  •**}  +  * 


77°  APPLIED  MECHANICS. 

When  ;r  =  36  -inches,  then  tan  a  =  — g-r — ; 

/*  —661507  ==  —732240  -f  615276  —  62208  4- 

482335 


c  =  — 


.'.    tana  =  -g-^j  —482335  —  20340*  4-  474.75** *sl. 

*  O        J 

When  x  =  84  inches, 

tana=  -^(-482335  -  1708560  +  3349836  -  790272) 

368669 
+~ET 
Portion  nearest  C, 


i  =  ry- 63660*  - 

368669 
When  ^r  :=  84  inches,  then  tan  a  —  — cy~  > 

/.  368669  =  5347440  —  176400  —  790272  -f- 

4012099 
•'•  r       ~^7^ 

.*.    tana  =  Tyj  —4012099  -h  63660*  —  25*2 *3>' 

When  x  =  120  inches, 

i  963101 

tana  =  -gj(— 4012099  -f-  7639200  —  360000  —  2304000)  =     gj 

Deflection. 

Portion  nearest  .#, 

»  =  -j-\  165600*  —  28I70*2  +  324.9*3  —  !*n. 

When  x  =:  36  inches, 

f  =  -^-,(165600  —  1014120  -f  421070  —  15552) (36) 
JiJ 

(443002)36  15948072 

El  El 


SLOPES. 


777 


Middle  portion, 

v  =  -^-\  —482335*  —  10170*'  4-  158.25*3  —  I** I  4.  c. 
•EJ  (  3     ) 

When  x  —  36  inches,  then  v  = ^j.       ; 

/.     -15948072  =  (-482335  -  366120  +  205092  —  15552)  -f  EIc, 

15289157 


•'•  V==^J\  ~I5289I57  -482335*-  10170**  -f  158.25*3-  i*»l. 

Greatest  deflection  occurs  in  the  middle  portion,  and  the 
point  is  given  by  the  equation. 

o  =  -482335  -  20340*  +  474-7S*2  —  i*3  =  o; 

.-.    *  =  71.4- 
Greatest  deflection  in  span  BC> 


FIG.  256. 
V  «= 

•^(-15289157-34438719-518462534-57602105-8662899) 

^52634923 

El 


i  20 

Hence,  putting  —  =        £.7  °,  we  obtain 
400  j^i 

EI=  175449743; 
therefore,  if  E  =  I4CXXXXD,  we  have 

/=  I25-3- 
If  b  =  3  inches,  h  =  8  inches. 


77$  APPLIED   MECHANICS. 


EXAMPLES   OF   CONTINUOUS   GIRDERS. 

i°.  Let  /  =  uniform  moment  of  inertia  of  girder. 

w  =  load  per  unit  of  length  uniformly  distributed. 
Find  expressions  for 

1,  the  bending-moment  over  each  support, 

2,  the  supporting-forces, 

3,  the  greatest  bending-moment, 

4,  the  slopes  at  the  supports, 

5,  the  greatest  deflection, 

in  each  of  the  following  cases  :  — 

(a)  Two  equal  spans,  length  /. 

(b)  Three  equal  spans,  length  /. 

(c)  Four  equal  spans,  length  /. 

(d)  Two  spans,  lengths  /,  and  12  respectively. 

(e)  Three  spans,  lengths  /,,  /2,  and  /3  respectively. 

(f)  Four  spans,  lengths  /„  /2,  /3,  and  /4  respectively. 

(g)  Two  equal  spans ;  loads  per  unit  of  length  on  each  span, 
wI  and  w2  respectively. 

(h]  Three  equal  spans  ;  loads  per  unit  of  length  on  each  span, 
wlt  zv2,  and  wz  respectively. 

2°.  Do  the  same  in  the  case  where  each  span  is  loaded  with 
a  centre  load  W,  and  has  no  distributed  load. 

3°.  Find  greatest  bending-moment  and  greatest  deflection 
for  a  continuous  girder  of  two  spans,  uniformly  loaded  on  these 
two  spans  with  load  w  per  unit  of  length,  and  which  overhang 
the  outer  supports  ;  the  overhanging  parts  having  lengths  /_0 
and  4  respectively,  and  the  same  distributed  load  per  unit  of 
length  on  the  overhanging  parts. 


EQUILIBRIUM  CURVES.  —  ARCHES  AND   DOMES.        779 


CHAPTER    IX. 

EQUILIBRIUM  CURVES.— ARCHES  AND  DOMES. 

§  248.  Loaded  Chain  or  Cord.  —  It  has  been  already  shown 
{§  126),  when  the  form  of  a  polygonal  frame  is  given,  that  the 
loads  must  be  adapted,  in  direction  and  magnitude,  to  that  form, 
or  else  the  frame  will  not  b£  stable.  The  same  is  true  of  a 
loaded  chain  or  cord,  which  would  be  realized  if  the  frame  were 
inverted. 

If  a  set  of  loads  be  applied  which  are  not  consistent  with 
the  equilibrium  of  the  frame  under  that  form,  it  will  change  its 
shape  until  it  assumes  a  form  which  is  in  equilibrium  under  the 
applied  loads. 

As  to  the  manner  of  finding  (when  a  sufficient  number  of 
conditions  are  given)  the  stresses  («) 

in  the  different  members,  etc.,  this 
Avas  sufficiently  explained  under  the 
head  of  "  Frames,"  and  will  not  be 
repeated  here,  as  the  figures  speak 
for  themselves. 

In  Fig.  257  the  polygon  fedcbaf 
is  the  force  polygon,  while  the  equilibrium  polygon  is  123456, 
an  open  polygon.  A  straight  line  joining  e  and  a  would  repre- 
sent the  resultant  of  the  loads. 


FIG.  257. 


APPLIED   MECHANICS. 


CHAIN   WITH   VERTICAL  LOADS. 

If  all  the  loads  are  vertical,  the  broken  line  edcba  becomes 
(«0  (*)        a   straight   line   and   vertical,    as 

shown  in  Fig.  258^.  Wheneverthe 
loads  are  concentrated  at  single 
points,  as  2,  3,  4,  5,  the  form  of 
the  chain  is  polygonal ;  and  when 
the  load  is  distributed,  it  becomes 
a  curve,  as  shown  in  Fig.  259. 


FIG.  258. 


CURVED  CHAIN   WITH   A  VERTICAL   DISTRIBUTED   LOAD. 

Given  the  form  of  the  chain  AOE  supported  at  A  and 
and  the  total  load  (a)  w 

upon  it  (be,  Fig. 
259^).  to  find  the 
distribution  of  the 
load  graphically. 
First  lay  off  be  to 
scale,  to  represent 
the  total  load  :  this 
is  balanced  by  the  two  supporting-forces  at  A  and  E  respec- 
tively, as  shown  in  the  figure.  Hence  draw  ca  parallel  to  the 
tangent  at  E,  and  ba  parallel  to  that  at  A,  and  we  have  the 
force  polygon  abca;  the  equilibrium  curve  being  the  chain  AOB 
itself.  Moreover,  if  the  lowest  point  of  the  chain  be  O,  then 
the  load  must  be  so  distributed  that  the  portion  between  O  and 
A  shall  be  balanced  by  the  tension  at  O  and  that  at  A,  and 
hence  that  its  resultant  shall  pass  through  the  intersection  of 
the  tangents  at  O  and  A.  Its  amount  will  be  found  by  drawing 
from  a  a  horizontal  line  ;  and  then  we  shall  have  ao  as  the  ten- 
sion at  o,  ab  as  the  tension  at  A,  and  bo  as  the  load  between  A 
and  O.  Hence  the  load  between  E  and  O  will  be  oc. 


LOADED   CHAIN  OR    CORD.  78 1 

Moreover,  the  load  between  O  and  any  point,  as  B,  will  be 
balanced  by  the  tension  at  O,  and  the  tension  at  B,  and  hence 
will  be  od,  where  ad  is  drawn  parallel  to  the  tangent  BD,  so 
that  the  load  between  B  and  E  will  be  dc ;  and  in  this  way  we 
see  that  we  can  find  the  tension  at  any  point  of  the  chain  by 
simply  drawing  a  line  from  a,  parallel  to  the  tangent  at  that 
point,  till  it  meets  the  load-line  be. 

It  is  to  be  observed,  that,  if  the  tension  at  any  point  of  the 
chain  be  resolved  into  horizontal  and  vertical  components,  the 
horizontal  component  will,  when  the  loads  are  all  vertical,  be  a 
constant,  and  the  vertical  component  will  be  equal  to  the  por- 
tion of  the  load  between  the  lowest  point  and  the  point  in 
question. 

If  we  assume  our  origin  at  O,  axis  of  x  horizontal  and  axis 
of  y  vertical,  and  let  the  co-ordinates  of  B  be  x  and  j,  and  if  w 
be  the  intensity  of  the  load  at  the  point  (x,  y\  we  shall  have,  for 
the  load  od  between  O  and  B, 

P  =  f^wdx; 

and,  since  the  angle  oad  —  angle  BDC,  we  shall  have 

^  =  —  =  —  =  — 
doc       DC~  oa~  H 

By  differentiation,  we  shall  have 

d 

d 

dx2          H 
or 

*y  =  ™, 

d*       H' 
and  this  is  the  equation  for  all  vertically  loaded  cords. 


782  APPLIED   MECHANICS. 

From  it  we  can  find  the  form  of  the  cord  to  suit  a  given 
distribution  of  the  load. 

§  249.  Chain  with  the  Load  Uni- 
formly Distributed  Horizontally.  — 
In  this  case  w  is  a  constant  ;  and  if 
we  assume  our  origin  at  the  lowest 
point  of  the  chain,  and  use  the  same 
notation  as  before,  we  shall  have 

d2y  _  w 


Hence,  integrating,  and  observing,  that,  when  x  =  p,  ~  =  o, 

we  have 

dy  _  wx  f 

~          ~ 


and  by  another  integration,  observing,  that,  when  x  =  o,  y  as  o, 

we  obtain 

w 


This  is  the  equation  of  a  parabola  ;  hence  a  chain  so  loaded 
assumes  a  parabolic  form. 

EXAMPLE  I.  —  Given  the  heights  of  the  piers  for  support- 
ing a  chain  so  loaded,  above  the  lowest  point  of  the  chain,  as 
8  and  18  feet  respectively,  the  span  being  100  feet,  to  find  the 
distance  of  the  lowest  point  from  the  foot  of  each  pier,  and 
the  equation  of  the  curve  assumed  by  the  chain. 

Solution.  —  If  (with  the  lowest  point  of  the  chain  as  origin) 
we  call  (jrn  y^  the  co-ordinates  of  the  top  of  the  first  pier,  and 
(^2»  72)  those  of  the  top  of  the  second  pier,  we  shall  have,  since 
y^  =  18  and^3  =  8,  and  since  we  must  have 


w 


CHAIN   WITH   UNIFORM  HORIZONTAL   LOAD.  783 


.     *'  _  i/18  -  *         .          _  3^         .  _  5^  . 

"      ~-  V    g    -   2  *x_   2*2  *a-   ^a, 

but 

^   -f-  X2  =    100          /.       f^2   =    100          /.       Xa  =   40,          Xj.   =   6O. 

Hence,  since  18  =  — 7>(6o)2 

a/          18  i 


2H         3600  200* 

therefore  equation  of  the  curve  is 


EXAMPLE  II.  —  Given  the  load  on  the  above  chain  as  4000 
Ibs.  per  foot  of  horizontal  length,  to  find  the  tension  at  the  low- 
est point,  also  that  at  each  end. 

Solution.    • 

w          i 
—  -  =  —  ,  w  —  4000, 

2H          200 

/.     2H  =  800000  /.     H  —  400000  Ibs. 

Moreover,  load  between  lowest  point  and  highest  pier  ±= 
60  X  4000  =  240000  Ibs. 

Therefore  tension  at  highest  pier  = 


^(240000)2  +  (400000)2  =  iooooy/(24)2  +  (4°)2 

=  10000^2176  =  466480  Ibs. 
Tension  at  lowest  pier  = 


^(i6oooo)2  H-  (400000)2  =  10000^/256  +  1600 

=  10000^1856  =  430813  Ibs. 


784  APPLIED   MECHANICS. 

EXAMPLE  III.  —  Given  the  span  of  the  chain  as  20  feet,  and 
its  length  as  25  feet,  the  two  points  of  support  being  on  the 
same  level,  to  find  the  position  of  the  lowest  point. 

§  250.  Catenary.  —  The  catenary  is  the  form  of  the  curve 
of  a  chain,  which,  being  of  uniform  section,  is  loaded  with  its 
own  weight  only,  i.e.,  with  a  load  uniformly  distributed  along 
the  length  of  the  chain. 

To  deduce  the  equation  of  the  catenary  :  if  we  assume  the 
origin,  as  before,  at  the  lowest  point  of  the  curve,  we  shaii 
have  still  the  general  equation 


w 


but  w  in  this  case  is  not  constant. 

If  we  let  Wi  =  the  load  per  unit  of   length  of   chain,  we 
shall  have 

ds 


hence 

d2y  _  wl  ds 
~dtf  ~  ~H~d^ 
Or,  if  we  let 

^=.  ± 
H       m 

a  constant, 

£?-.!*  d\ 

da*  ~  m  dx9  U> 

which  is  the  differential  equation  of  the  catenary  ;  and  we  only 
need  to  integrate  it  to  obtain  the  equation  itself. 
To  do  this,  we  have 

dzy 


m 


•HE  CATENARY. 


therefore,  integrating,  and  observing,  that,  when  x  ==.  o,  —  =  o, 
we  shall  have 


• 


dy        \l  *.         -£\  ml  *.         . 

.-.     -f  =  -(ent  -  e  **  \          .-.    y  =  —(e™  +  e  >*\  +  c. 
dx       2\  I  2\  / 

But,  when  ;r  =  o,^  =  o      .'.     c  =  —m;  hence  the  equation  is 

y  =  f(^  +  e~™)  -  m>  (s) 


and  this  is  the  equation  of  the  catenary 
when  the  origin  is  taken  at  O,  the  low- 
est point  of  the  chain. 

If  it  be  transferred  to  Ol}  where  OOl 
^-m,  the  equation  becomes  (by  putting 
—  m) 


FIG.  261. 


ml  *    ,      _^\ 
=  —  (  e**  -H  e   m\. 


(4) 


This  is  the  most  common  form  of  the  equation  to  the  cate- 
nary, the  origin  being  taken,  at  a  distance  below  the  lowest 

TT 

point  of   the  curve  equal  to  m  =  — ,  the  horizontal  tension 


w, 


divided  by  the  weight  per  unit  of  length  of  chain. 


APPLIED   MECHANICS. 


To  find  x  in  terms  of  y,  we  have 


Solving,  we  have 


m  m 


To  find  the  length  of  the  rope  :  from  the  equation 


m    £.  * 

y  —  — 


we  obtain 

dy       i  /  £.        _£ 
_Z  =  _^w  —  <f  w 

dx 


.    -         -        4-  ^~  (6) 

dx 


To  find  the  area  OO^A^B,  we  have 
Area  =     V  =  *(<*  +  <-*<**  =        <    -  *-«•  (« 


TRANSFORMED   CATENARY.  787 

But 

arc  OB  =  -e%  -  *~£   -  // 


hence,area  OO^A^B  =  ms. 

This  shows,  that,  if  the  load  should  be  distributed  in  such  a 
way  as  to  be  like  a  uniformly  thick  sheet  of  metal,  having  for 
one  side  the  catenary  and  for  the  other  the  straight  line  O^Alt 
the  equilibrium  curve  would  be  a  catenary. 

It  may  be  convenient  to  have  the  development  of  e"*  .and 

JC_ 

e~my  hence  they  will  be  written  here  :  — 


m 


m 


EXAMPLE  I.  —  Given  a  rope  90  feet  long,  spanning  a  hori- 
zontal distance  of  75  feet  ;  find  the  equation  of  the  catenary, 
the  sag  of  the  rope,  and  the  inclination  of  the  rope  at  each 
support,  supposing  these  to  be  on  the  same  level. 

§  251.  Transformed  Catenary.  —  We  have  just  seen  that 
the  catenary  is  the  form  of  chain  suited  to  a  load  which  may 
be  represented  by  a  uniformly  thick  sheet  of  metal,  with  a  hori- 
zontal extrados,  provided  the  distance  OOI  is  equal  to  m,  a  defi- 
nite quantity.  A  more  general  case,  however,  would  be  that  of 
a  chain  loaded  with  a  load  which  might  be  represented  by  a 
uniformly  thick  sheet  of  metal,  where  the  length  OOl  is  any 
given  quantity  whatever.  A  chain  so  loaded  is  called  a  trans- 
formed catenary,  and  the  catenary  itself  becomes  a  particular 
case  of  the  transformed  catenary. 

We  may  deduce  its  equation  as  follows  :  — 


APPLIED   MECHANICS. 


Let   the  chain  be  represented   by  ACB,  and  let  it  be  so 
loaded  that  the  load  on  CD  is  repre- 
sented by  w  times  area  OCDE,  so  that 
B      w  =  weight  per  unit  of  area ;  then  we 
shall  have,  for  this  load, 


p  = 


Hence,  from  what  we  have  already  seen, 
dy        P        w 


ydx 


d*y       w 


fyd2), 

• '     dx  dx2 


w  dy 
-fry-j- 
Jf  dx 


integrating,  we  have 

(dy\2  _  w 
\Tx)    "-  H 


/    But,  when  -j-  =  O,  y  =  a  ; 


Idy  \2 

•••  (i) 


w 


TT 

Or,  if  we  write,  for  brevity,  —  =  m2,  we  have 

y2  —  a2 


dx 


m2 


dy         i 

•"•     ^T  —  ~^ 

dx       m 


dy_ 

Vy2  -  a2 


+  «r--o-s 


LINEAR  ARCH. 


But,  when  x  =  oty  =  a  ; 


2.T 


which  is  the  equation  of  the  transformed  catenary.  This 
becomes  the  catenary  itself  whenever  a  =  w. 

EXAMPLE.  —  Given  a  chain  loaded  so  that  the  load  on  CD  is 
proportional  to  the  area  OEDC.  Let  OC  —  $  feet,  BF  =  8  feet, 
(9T7  =  4  feet  ;  weight  per  unit  of  area  —  80  Ibs.  Find  the 
equation  of  the  transformed  catenary,  also  the  tension  at  C  and 
that  at  B. 

§  252.  Linear  Arch  --  In  all  the  preceding  cases,  the  chain 
or  cord  is  called  upon  to  resist  a  tensile  stress  arising  from  a 
load  that  is  hung  upon  it.  If,  now,  the  cord  be  inverted,  we 
have  the  proper  equilibrium  curve  for  a  load  placed  upon  it,  dis- 
tributed in  the  same  manner  as  before  ;  only  in  this  latter  case 
the  cord  would  be  subjected  to  direct  compression  throughout 
its  whole  extent.  The  equilibrium  curve  is,  then,  sometimes 
called  a  linear  arch.  The  general  equation  of  the  equilibrium 
curve  remains  just  as  before, 


the  axes  being  so  chosen  that  OX  is  horizontal  and  OY  verti- 
cal. 

Thus,  if  it  were  required  to  find  the  form  of  the  equilibrium 
curve  or  linear  arch,  with  the  upper  boundary  of  the  loading 
horizontal,  we  should  obtain  a  transformed  catenary. 


79°  APPLIED   MECHANICS. 

§  253.  Arches.  —  In  the  case  of  arches  composed  of  a  series 
of  blocks,  as  in  stone  or  brick  arches,  the  mathematical  treat- 
ment generally  used  for  determining  the  proper  form  and 
proportions  of  the  arch  has  been  quite  different  from  that  used 
for  the  determination  of  the  proper  form  and  proportions  of 
the  iron  arch,  whether  made  in  one  piece,  or  two  pieces  hinged 
together,  or  of  a  lattice. 

In  the  case  of  the  iron  arch,  the  treatment  involves  neces- 
sarily a  determination  of  the  stresses  acting  in  all  its  parts,  and 
an  adaptation  of  its  form  and  dimensions  to  the  load,  so  that  at 
no  point  shall  the  stress  exceed  the  working-strength  of  the 
material. 

In  the  case  of  the  stone  arch,  it  is  still  a  question  under 
discussion  whether  it  would  not  be  best  to  adopt  the  same 
method,  although  it  would  lead  to  a  great  deal  of  complexity,  on 
account  of  the  joints. 

Nevertheless,  the  question  usually  raised  is  one  merely  of 
stability  ;  i.e.,  as  to  the  proper  form  and  dimensions  to  pre- 
vent, not  the  crushing  of  the  stone,  though  this  must  also  be 
taken  into  account  if  there  is  any  danger  of  exceeding  it,  but 
more  especially  the  overturning  about  some  of  the  joints. 

The  question  of  the  stability  of  the  stone  arch  may  present 
itself  in  either  of  the  two  following  ways  :  — 

i°.  Given  the  arch  and  its  load, -to  determine  whether  it  is 
stable  or  not. 

2°.  Given  the  distribution  of  the  load,  to  determine  the 
suitable  equilibrium  curve,  and  hence  the  form  of  arch,  suited 
to  bear  the  given  load  with  the  greatest  economy  of  material. 

§  254.  Modes  of  giving  Way  of  Stone  Arches.  —  An  arch 
may  yield,  (i°)  by  the  crushing  of  the  stone,  (2°)  by  sliding  of 
the  joints,  (3°)  by  overturning  around  a  joint.  The  following 
figures  show  the  modes  of  giving  way  of  an  arch  by  the  last 
two  methods.  The  first  two  show  the  dislocation  of  the  arch 
by  the  slipping  of  the  voussoirs.  In  the  former  case  the 


FRICTION. 


791 


hatmches  of  the  arch  slide  out,  and  the  crown  slips  down  ;  in 
the  other  case  the  reverse  happens.  The  second  two  figures 
show  the  two  methods  by  which  an  arch  may  give  way  by 
rotation  of  the  voussoirs  around  the  joints. 


FIG.  263. 


FIG.  264. 


FIG.  265. 


FIG.  266. 


Before  proceeding  farther  with  the  problem  of  the  arch,  two 
or  three  matters  of  a  more  general  nature  will  be  treated, 
which  will  be  necessary  in  its  discussion. 

§  255.  Friction. — Let  AB  be  a  plane  inclined  to  the  hori- 
zon at  an  angle  0.  Let  D  be  a  body  resting 
on  the  plane,  of  weight  DG  =  W.  Resolve 
W  into  two  components,  DE  and  DF  respec-  CL 
tively,  perpendicular  and  parallel  to  the 
plane.  The  component  DE  =  WcosO  is 
entirely  neutralized  by  the  re-action  of  the  plane ;  while  DF 
=.  Wsin.  0,  on  the  other  hand,  is  the  only  force  tending  to  make 
the  body  slide  down  the  plane.  It  is  an  experimental  fact,  that 
when  the  angle  0  is  less  than  a  certain  angle  </>,  called  the 
angle  of  repose,  the  body  does  not  slide ;  when  0  =  </>,  the  body 
is  just  on  the  point  of  sliding ;  and  when  0  is  greater  than  <£, 
the  body  slides  down  the  plane  with  an  accelerated  motion, 
showing  that  in  this  case  an  unbalanced  force  is  acting.  This 


792  APPLIED   MECHANICS. 

angle  <£  depends  upon  the  nature  of  the  material  of  the  plane 
and  of  the  body,  and  on  the  nature  of  the  surfaces.  Hence, 
in  the  first  and  second  cases,  the  friction  actually  developed 
by  the  normal  pressure  DE  just  balances  the  tangential  com- 
ponent DF;  whereas,  in  the  third  case,  when  the  angle  of 
inclination  of  the  plane  to  the  horizon  is  greater  than  <£,  the 
tangential  component  DF  is  only  partially  balanced  by  the 
friction. 

Let  ab  be  the  plane  when  inclined  to  the  horizon  at  an 
angle  <f>.  The  body  is  then  just  on  the 
point  of  sliding,  hence  the  component 
df  —  Wsmcj>  is  just  equal  to  the  fric- 


«^j  tion  developed  between  the  two  surfaces. 

FIG.  268.  Moreover,    if    we    represent   by   TV  the 

normal  pressure  de  =  IV COB  <£  on  the  plane,  we  shall  have 

df  =  TV  tan  <£. 

Now,  it  is  an  experimental  fact,  that  the  friction  developed 
between  two  given  surfaces  depends  only  on  the  normal  press- 
ure, i.e.,  that  the  friction  bears  a  constant  ratio  to  the  normal 
pressure ;  and  since,  in  this  case,  the  friction  just  balances  the 
tangential  component  df  =  TV  tan  <£,  the  friction  due  to  the 
normal  pressure  TV  is 

TV  tan  <£. 

Now,  it  makes  no  difference  what  be  the  position  of  the 
plane  surface :  if  a  normal  pressure  TV  be  exerted,  the  friction 
that  is  capable  of  being  exerted  to  resist  any  force  F  tangential 
to  the  plane,  tending  to  make  the  bodies  slide  upon  each  other, 
is  TV  tan  <£  /  and  if  the  force  F  is  greater  than  TV  tan  <£,  the  bodies 
will  slide,  but  if  Fis  less  than  TV  tan  <£,  they  will  not  slide.  The 
quantity  tan  <£  is  called  the  co-efficient  of  friction,  and  will  be 
denoted  by  f. 


STABILITY  OF  BUTTRESS  ABOUT  A    PLANE  JOINT.     793 

From  the  preceding  it  is  evident,  that,  if  the  resultant  press- 
ure on  the  body  makes  with  the  normal  to  the  plane  an  angle 
less  than  the  angle  of  repose,  the  sliding  will  not  take  place ; 
whereas,  if  the  resultant  force  makes  with  the  normal  to  that 
plane  an  angle  greater  than  the  angle  of  repose,  the  body  will 
slide. 

§  256.  Stability  of  Position. — To  determine  under  what 
conditions  the  stability  of  the  block 
DGHFis  secure  against  turning  around  A 
the  edge  D:  if  the  resultant  of  the 
weight  of  the  block  and  the  pressure 
thereon  pass  outside  the  edge  D,  as  ORlt  ( 
then  the  block  will  overturn  ;  the  mo- 
ment of  the  couple  tending  to  overturn  it 
being  OR,  X  DE.  If,  on  the  other 
hand,  it  pass  within  the  edge,  as  OR2,  the  block  will  not  over- 
turn, since  the  force  has  a  tendency  to  turn  it  the  opposite 
way  around  D.  Hence,  in  order  that  a  block  may  not  overturn 
around  an  edge  at  a  plane  joint,  the  resultant  pressure  must 
cut  the  joint  within  the  joint  itself. 

In  any  structure  composed  of  blocks  united  at  plane  joints, 
we  must  have  both  stability  of  position  and  stability  of  friction 
at  each  joint,  in  order  that  the  structure  may  not  give  way. 

§257.  Stability  of  a  Buttress  about  a  Plane  Joint. — 
Let  DCEF  be  a  vertical  section  of  a  buttress,  against  which 
a  strut  rib  or  piece  of  framework  abuts,  exerting  a  thrust 
P  =  ZX  =  OR.  In  order  that  the  buttress  may  not  give 
way,  it  must  fulfil  the  conditions  of  stability  at  each  joint.  Let 
AB  be  a  joint.  Should  several  pressures  act  against  the  but- 
tress, the  force  P  in  the  line  ZO  may  be  taken  to  represent 
the  resultant  of  all  the  thrusts  which  act  on  the  buttress  above 
the  joint  AB.  Let  G  be  the  centre  of  gravity  of  the  part 
ABEF,  and  let  W  —  OL  be  the  weight  of  that  part  of  the 
buttress.  Let  0  be  the  point  of  intersection  of  the  line  of 


704 


APPLIED    MECHANICS. 


direction  of  the  thrust,  and  of  the  weight  W.  Draw  the  paral- 
lelogram ORNL.  Then  will  ON  be  the  resultant  pressure  on 
the  joint  AB :  and  the  conditions  of  stability  require  that  the 
resultant  pressure  should  cut  the  joint 
AB  at  some  point  between  A  and  B,  and 
that  its  line  of  direction  should  make 
with  the  normal  to  AB  an  angle  less 
than  the  angle  of  repose,  <£;  and,  in 
order  that  the  buttress  may  not  give  way, 
these  conditions  must  be  fulfilled  at  each 
and  every  joint. 

Another  way  of  expressing  this  con- 
dition is  as  follows  :  The  force  tending 
to  overturn  the  upper  part  of  the  but- 
tress around  A  is  the  force  F  =  OR ; 
and  its  moment  around  A  is  F(Ap)  =  Fp 
if  we  let  Ap  =  p,  whereas  the  moment  of  the  weight  which 
resists  this  is  W(AS)  =  Wq  if  we  let  AS  =  q.  Now,  when 
ON  passes  through  A,  we  have  Fp  —  Wq ;  when  ON  passes 
inside  of  A,  we  have  Wq  >  Fp  ;  when  ON  passes  outside  of  A, 
we  have  Wq  <  Fp.  Hence  the  conditions  of  stability  require 
that 

Wq^.Fp        or        Fp^Wq. 

EXAMPLE.  —  Given  a  rectangular  buttress 
8  feet  high,  i  foot  wide,  and  4  feet  thick ;  the 
weight  of  the  material  being  100  Ibs.  per 
cubic  foot,  the  buttress  being  composed  of  8 
rectangular  blocks  1X4X1  foot.  On  this 
buttress  is  a  load  of  500  Ibs.,  whose  weight 
acts  through  K,  where  OK  =  3  feet.  Find 
the  greatest  horizontal  pressure  P  that  can  be  applied  along 
the  line  OK,  consistent  with  stability,  against  overturning 
around  each  of  the  edges  a,  bt  c,  d,  e,  ft  gt  h. 


FIG.  271. 


LINE    OF  RESISTANCE   IN  A    STONE   ARCH.  795 


Solution.  -  -  The    weight   of   each    block   will    be   400   Ibs, 
Hence  we  shall  have  the  following  equations  :  — 


1500  4-  400  x  2 
Stability  about  a,  max  P  —  -          — —      —    =  2300. 

iqoo  4-  800  x  2 

=  1550. 


2 
I5OO    -f-    I2OO    X     2 


-  1300. 


-f    I60O  X    2 

"     d,       «    =  -        —       -  =  1175- 

„      _  '500  +  2000  x  ,  =  iiop 

«  /,    •<  =  I5°° + 1400  x  2  =  1050. 


X 


The  least  of  these  being  987  Ibs.,  it  follows  that  the  great- 
est pressure  consistent  with  stability  against  overturning  is 
987  Ibs. 

§  258.  Line  of  Resistance  in  a  Stone  Arch.  —  In  order 
to  solve  any  problem  involving  the  stability  of  a  stone  arch,  it 
is  necessary  that  the  student  should  be  able  to  draw  a  line  of 
resistance.  To  make  plain  the  meaning  of  the  term,  the  follow- 
ing solution  of  an  example  is  given.  The  method  of  drawing 
the  line  of  resistance  employed  in  this  solution  is  given  purely 
for  purposes  of  illustration,  and  is  not  recommended  for  use- in 
practice,  as  a  suitable  method  will  be  given  later. 


APPLIED    MECHANICS. 


EXAMPLE. — Given  three  blocks  of  stone  of  the  form  shown 

in  the  figure  (Fig.  272),  their 
common  thickness  (perpendic- 
ular to  the  plane  of  the  paper) 
being  such  that  the  weight 
per  square  inch  of  area  (in  the 
plane  of  the  paper)  is  just  one 
pound. 

Given    AC    =    13     inches, 
EC    =     8    inches.      Suppose 
these  three  blocks  to  be  kept 
from    overturning  by    a    hori- 
zontal   force    applied    at    the 
middle  of  DE.     Find  the  least 
value  of  this  horizontal  force  consistent  with  stability  about 
the  inner  joints,  also  its  greatest  value  consistent  with  stabil- 
ity about  the  outer  joints. 
Solution. 

BK  =  16  sin  15°  =  4.14112. 
AH '=  26  sin  15°  =  6.72932. 


N2 


FIG.  272. 


3((i3)2-(S)2 


.  _  I0 

/r,\->  r  —  IO< 7* 


Altitude  of  each  trapezoid  —  5  cos  15°        =     4.8296. 

-sin  30°          =  26.25  sq.  in. 
=  26.25  Ibs. 


Area  of  each  trapezoid  = 
Weight  of  each  stone 


GG2  —  8  sin  30°  —  10.7  cos  15°  sin  15°  =  1.325. 

=  8  cos  30°  —  10.7  cos  15°  sin  15°  =  4.253. 

=  10.7  cos  15°  cos  45°  —  8  cos  30°  =  0.380. 

JBN2  =  8  —  10. 7  cos  15°  sin  15°  =  5-325- 

—  8  —  10.7  cos  15°  cos  45°  =  0.692. 

r4  =  10.7  cos2 15°  —  8  =  1.983- 

=  13  cos  30°  •-  10.7  cos  15°  sin  15°  =  8.583. 

=  13  cos  30°  --  10.7  cos  15°  cos  45°  =  3-95°- 


LINE    OF  RESISTANCE  IN  A   STONE   ARCH. 

AN2   =  13  —  10.7  cos  15°  sin  15°  =  10.325. 

ANZ   =  13  —  10.7  cos  15°  cos  45°  =  5.692. 

AJV4   =  13  —  io.7cos2i5°  =  3.017. 

G,M  =  10.5  -  8  cos  30°  =  3.572. 

Ki  M  —  10.5  —  8  sin  30°  =  6.500. 

CM   —  10.5  =  10.500. 

Hi  M  =  10.5  —  1  3  sin  30°  =  4.000. 

Let  us  represent  the  thrust  at  Mby  T.  Then,  to  find  what 
is  the  thrust  required  to  produce  equilibrium  about  G,  we  take 
moments  about  G,  and  likewise  for  the  other  joints.  We  may 
proceed  as  follows  :  — 

INNER  JOINTS. 

Stability  about  G, 

T(G,M}  =  (26.25)  (GG2) 
or 

7X3.572)  =  (26.25)  (1.325)  A     T=     9.74. 

Stability  about  K, 

T(K,M)  =  (26.25)  (KK2  -  KK,) 
or 

7X6.500)  =  (26.25)  (4.253  -  0.380)  .•.     T=  15.64.. 

Stability  about  B, 

T(CM)    =  (26.25)  (BN2  +  UN,  -J5N4)     .%     T=    1-0.08 


OUTER  JOINTS. 

Stability  about  H, 

T^M)  ==  (26.2$)(HH2  +  HH,)  /.     T=  82.25. 

Stability  about  A, 

T(CM)    =  (26.25)  (AN2  +  AN,  +  AN<)      .-.     T=  47-59 

It  is  plain,  therefore,  that,  in  order  to  have  equilibrium,  the 


A  rn. 


'<  Y/.-I  .\vc  \y. 


thrust  at  M  must  bo  between    15.64  Ibs.  and  47.59  Ibs.  :  for, 
it  it  is  less  than    15.64  Ibs.,  the  areh  will  turn  about  an  inner 

joint  ;  and  if  it  is  greater  than 
47.59  Ibs.,  it  will  turn  around 
an  outer  joint. 

If,  now,  we  draw  through 
AF  a  horizontal  line  to  meet 
the  vortical  drawn  through  the 
centre  of  gravity  of  the  first 
stone,  and  lay  off  aft  =-  15.64, 
and  «y  --  26.25,  then  will  the 
resultant  of  this  thrust  aft  and 
the  weight  of  the  first  stone  <»y 
be  «8 ;  this  being  the  resultant 
pressure  on  the  joint  /-Y7,  its 
point  of  application  being  «. 
Next,  prolong  this  line  u£  to 
meet  the  vertical  through  the 
centre  of  gravity  of  the  second 
stone,  and  combine  a 8  with  the 
weight  of  the  second  stone, 
thus  obtaining,  as  resultant 
pressure  on  the  joint  A7A  the 
force  £>/,  whose  point  of  appli- 
cation is  at  A*.  Compounding, 
now,  ;>;  with  the  weight  of  the 
third  stone,  we  obtain,  as  final 
resultant  pressure  on  .-J/\  the 
force  A,«  applied  at  p.  Xow, 
joining  JAA'p  by  a  broken  line,  we  have  the  Line  </  Rcsistaticc 
corresponding  to  the  thrust  15.64,  or  the  minimum  horizontal 
tluust  at  AF.  If,  now,  we  construct  a  line  of  resistance  with 
47  59  Ibs.,  we  obtain  the  line  JAr<^-J,  corresponding  to  maximum 
horizontal  thrust  at  AF. 


SYMMh'.TKlCAI.    DISTRIBUTION  Oh'    Till'.    /.(MA  799 

II  the  arch  is  in  equilibrium,  rind  if  the  horizontal  thrust  is 
applied  at  J/,  it  is  plain  that  the  actual  thrust  would  cither  he 
one  of  these  two  or  else  somewhere  between  these  two,  and 
hence,  that,  if  the  requisite  thrust  is  furnished  at  M  to  keep 
the  arch  in  equilibrium,  the  true  line  ol  resistance  cannot  lie 
outside  of  these  two  ;  viz.,  the  line  corresponding  to  maximum 
and  that  corresponding  to  minimum  horizontal  thrust  at  /?/. 

If  the  separate  stones  supported  loads,  it  would  be  neces- 
sary to  take  into  account  these  loads,  in  addition  to  the  weights 
of  the  stones,  in  determining  the  horizontal  thrust,  and  drawing 
the  lines  of  resistance. 

§  259.  Arches  with  Symmetrical  Distribution  oi  the 
Load. —  Before  considering  the  conditions  of 
stability  of  an  arch,  we  shall  proceed  to  some 
propositions  about  lines  of  resistance  corre- 
sponding to  maximum  and  minimum  horizon- 
tal thrust.  If,  in  an  arch,  we  draw  a  line  of 
resistance  ///>'  through  the  point  //  ol  the 
crown,  and  then,  by  changing  the  horizontal 
thrust,  we  change  the  line  of  resistance  con- 
tinuously till  it  touches  the  extrados  of  the  arch  at.  (/',  we' 
shall  evidently  have,  in  the  line  //f /?',  a  line  of  resistance' 
which  has  the  greatest  horizontal  thrust  of  any  line  that,  passes 
through  A,  and  lies  wholly  within  the  arch-ring.  If,  on  the 
other  hand,  we  decrease  gradually  the  horizontal  thrust  until 
the  line  touches  the  intrados  at  //,  then  we  have  in  this  line 
the  line  of  minimum  horizontal  thrust  that  passes  through  //. 
JJy  lowering  the  point  A,  however,  and  keeping  the  point  ft  he- 
same,  we;  should  obtain  new  lines  of  resistance  with  greater 
and  greater  horizontal  thrust  ;  the  greatest  being  attained  when 
the  line  comes  to  have  one  point  in  common  with  the  intrados. 
Hence  a  line  of  maximum  horizontal  thrust  will  have  one  point 
in  common  with  the  extrados  and  one  point  in  common  with 
the  intrados,  the  latter  being  above  the  former. 


80O  APPLIED   MECHANICS. 

On  the  other  hand,  by  retaining  the  point  D'  the  same,  and 
raising  the  point  A,  we  should  decrease  the  horizontal  thrust, 
and  thus  obtain  lines  of  resistance  with  less  and  less  horizontal 
thrust ;  the  least  being  attained  when  the  line  of  resistance 
comes  to  have  a  point  in  common  with  the  extrados.  Hence 
the  minimum  line  of  resistance  has  a  point  in  common  with  the 
extrados  and  one  in  common  with  the  intrados,  the  latter  being 
below  the  former. 

These  cases  are  exhibited  in  the  following  figures  :  — 

,  minimum 


minimum 
maximum 


Fie.  275. 


§  260.  Conditions  of  Stability.  —  The  question  of  the  sta- 
bility of  an  arch  must  depend  upon  the  position  of  its  true 
line  of  resistance.  If  this  true  line  of  resistance  lies  within 
the  arch-ring,  the  arch  will  be  stable  provided  the  material 
of  which  it  is  made  is  incompressible.  If  this  is  not  the  case, 
the  stability  of  the  arch  will  depend  upon  how  near  the  true 
line  of  resistance  approaches  the  edge  of  the  joints ;  for  the 
nearer  it  approaches  the  edge  of  a  joint,  the  greater  the  inten- 
sity of  the  compressive  stress  at  that  joint,  and  the  greater  the 
danger  that  the  crushing-strength  of  the  stone  will  be  exceeded 
at  that  joint.  Thus,  if  the  true  line  of  resistance  cuts  any 
given  joint  at  its  centre  of  gravity,  the  stress  upon  that  joint 
will  be  uniformly  distributed  over  the  joint.  If,  however,  it 
cuts  the  joint  to  one  side  of  its  centre  of  gravity,  the  intensity 
of  the  stress  will  be  greater  on  that  side  than  on  the  opposite 
side ;  and,  if  it  is  carried  far  enough  to  one  side,  we  may  even 
have  tension  on  the  other  side. 


CRITERION  OF  SAFETY  FOR  AN  ARCH. 


801 


§  261.  Criterion  of  Safety  for  an  Arch.  —There  are  two 
criteria  of  safety  for  an  arch,  that  have  been  used  :  — 

i°.  That  the  line  of  resistance  should  cut  each  joint  within 
such  limits  that  the  crushing-strength  of  the  stone  should  not 
be  exceeded  by  the  stress  on  any  part  of  the  joint. 

2°.  That,  inasmuch  as  the  joint  is  not  suited  to  bear  tension 
at  any  point,  there  should  be  no  tension  to  resist. 

The  distribution  of  the  stress  is  assumed  to  be  uniformly 
varying  from  some  line  in  the  plane  of  the  joint.  The  three 
following  figures  will,  on  this  supposition,  represent  the  three 
cases : — 

i°.  When  the  stress  is  wholly  compression. 

2°.  When  the  stress  becomes  zero  at  the  edge  B. 

3°.  When  the  stress  becomes  negative  or  tensile  at  B. 

R, 


\ 


R  O 

FIG.  278. 


\J, 


FIG.  279. 


In  all  three  figures,  AB  represents  the  joint  which  is  as- 
sumed to  be  rectangular  in  section,  AD  represents  the  intensity 
of  the  stress  at  A,  and  BE  that  of  the  stress  at  B ;  while  R  repre- 
sents the  point  of  application  of  the  resultant  stress,  RR^  rep- 
resenting that  resultant. 

PROPOSITION.  —  If  the  stress  on  a  rectangular  joint  vary 
uniformly  from  a  line  parallel  to  one  edge,  the  condition  that 
there  shall  be  no  tension  on  any  part  requires  that  the  result- 
ant of  the  compressive  stress  shall  be  limited  to  the  middle 
third  of  the  joint. 

PROOF.  —  Let  AB  (Fig.  278)  represent  the  projection  of 
the  ioint  on  the  plane  of  the  paper.  It  is  assumed  that  the 


802  APPLIED   MECHANICS. 

stress  is  uniformly  varying  ;  and,  if  there  is  to  be  no  tension 
anywhere,  the  intensity  at  one  edge  must  not  have  a  value  less 
than  zero,  hence  at  the  limiting  case  the  value  must  be  zero  ; 
hence  this  limiting  case  is  correctly  represented  by  the  figure, 
and  the  resultant  of  the  compression  will  be  for  this  case  at  the 
centre  of  stress.  Thus,  if  AD  represent  the  greatest  intensity 
of  the  stress,  then  "we  shall  have,  if  B  be  the  origin  and  BA  the 
axis  of  x,  if  the  axis  of  y  be  perpendicular  to  AB  at  B,  and  if 
we  let  a  =  intensity  of  stress  at  a  unit's  distance  from  B,  that 
RR,  =  aSSxdxdy,  and  (BR)  (RRZ)  =  affx'dxdy; 


f  fx*dxd\          3 

727?  J  J  _  —   _ 

*•*•"•    =    ~~~  -  '  —  ~    —    777 

bh* 


if  b  =  breadth,  and  h  —  BA  =  height  of  rectangle. 

Hence,  if  the  resultant  of  the  compression  be  nearer  A  than 
Ry  there  will  be  tension  at  B  ;  and,  on  the  other  hand,  if  it  be 
nearer  B  than  \h,  there  will  be  tension  at  A.  Hence  follows 
the  proposition  as  already  stated. 

While  the  above  is  probably  the  condition  most  generally 
used  to  determine  the  stability  of  an  arch,  at  the  same  time,  if 
there  is  any  danger  that  the  intensity  of  the  stress  at  any  part 
of  any  joint  may  exceed  the  working  compressive  strength  of 
the  stone,  this  ought  to  be  examined,  and  hence  a  formula  by 
which  it  may  be  done  will  be  deduced. 

Let  AB  (Fig.  279)  be  the  joint,  and  let,  as  before,  b  be  its 
breadth,  and  h  =  AB  =  depth  ;  then,  suppose  the  pressure  to 
be  uniformly  varying,  DA  =f=  the  working-strength  per  unit 
of  area  =  greatest  ^allowable  intensity  of  compression  ;  then  the 
entire  stress  on  the  joint  will  be  represented  by  the  triangle 
A  CD,  for  the  joint  is  incapable  of  resisting  tension. 
Hence 

AR  =  \AC  :. 


POSITION  OF   THE    TRUE   LINE   OF  RESISTANCE.        803 


but 


and  this  is  the  least  distance  from  the  outer  edge  at  which  the 
resultant  should  cut  the  joint. 

We  thus  obtain,  in  terms  of  the  pressure  on  any  joint,  and 
of  the  working-strength  of  the  material,  the  limits  within  which 
the  line  of  resistance  should  pass,  in  order  that  the  working- 
strength  of  the  stone  may  not  be  exceeded. 

§  262.  Position  of  the  True  Line  of  Resistance.  —  The 
question  of  the  most  probable  position  of  the  true  line  of 
resistance  involves  the  discussion  of  the  properties  of  the 
elastic  arch.  This  discussion  will  be  given  later  ;  but,  for  the 
present,  the  statement  only  of  the  following  proposition,  due  to 
Dr.  Winkler,  will  be  given  :  — 

"For  an  arch  of  constant  section,  that  line  of  resistance  is 
approximately  the  true  one  which  lies  nearest  to  the  axis  of  ttit 
arch-ring,  as  determined  by  the  method  of  least  squares." 

From  this  it  will  follow  :  — 

i°.  That,  if  a  line  of  resistance  can  be  drawn  in  the  arch- 
ring,  then  the  true  line  of  resistance  will  lie  in  the  arch-ring  ; 
and 

2°.  That,  if  a  line  of  resistance  can  be  drawn  within  the 
middle  third  of  the  arch-ring,  then  the  true  line  of  resistance 
will  lie  in  the  middle  third. 

'But,  before  proving  this  proposition,  the  proposition  will  be 
used,  and  the  method  explained,  for  determining  whether  a  line 
of  resistance  can  be  drawn  within  the  arch-ring  :  for,  if  it  can, 
then  the  true  line  of  resistance  must  lie  within  the  arch-ring  ; 
and  if  no  line  of  resistance  can  be  drawn  within  the  arch-ring, 
then  the  true  line  of  resistance  cannot  pass  within  the  arch'« 
ring,  and  the  arch  would  necessarily  be  unstable,  even  if  the 
materials  were  incompressible. 

By  following  the  same  method,  we  could  determine  whether 


804  APPLIED  MECHANICS. 

it  was  possible  to  draw  a  line  of  resistance  within  the  middle 
third  of  the  arch-ring ;  and,  if  this  is  found  to  be  possible,  we 
should  know  that  the  true  line  of  resistance  will  pass  within 
the  middle  third  of  the  arch-ring. 

Hence  our  most  usual  criterion  of  the  stability  of  a  stone 
arch  is,  whether  a  line  of  resistance  can  be  passed  within  the 
middle  third  of  the  arch-ring. 

If  the  condition  be  used,  that  the  working-strength  of  the 
stone  for  compression  be  not  exceeded,  then,  instead  of  the 
middle  third,  we  shall  have  some  other  limits. 

In  what  follows,  an  explanation  will  be  given  of  Dr.  Scheff- 
ler's  method  (that  most  commonly  employed)  of  determining 
whether  a  line  of  resistance  can  be  drawn  within  the  arch-ring, 
inasmuch  as  the  same  method  can  be  employed  to  determine 
whether  such  a  line  can  be  drawn  within  the  middle  third  or 
within  any  other  given  limits. 

§  263.  Preliminary  Proposition  referring  to  Arches  Sym- 
metrical in  Form  and  Loading.  —  An  arch  and  its  load  being 
given,  a  line  of  resistance  can  always  be  made  to  pass  through 
any  two  given  points ;  hence,  if  any  two  points  of  a  line  of 
resistance  are  given,  the  line  is  determined. 

Proof.  —  Let  the  arch  be  that  shown  in  Fig.  281  ;  and  let  us 
consider  first  the  special  case  when  the  two  given  points  are  A, 
the  top  of  the  crown-joint,  and  G4,  the  foot  of  the  springing- 
joint.  In  this  case,  the  only  quantity  to  be  determined  is  the 
thrust  at  A.  Let  this  thrust  be  denoted  by  T;  let  P  be  the 
total  weight  of  the  half-arch  and  its  load ;  let  a  be  the  perpen- 
dicular distance  of  the  point  G4  from  a  vertical  line  through  the 
centre  of  gravity  of  the  entire  half-arch  and  its  load ;  let  h  be 
the  vertical  depth  of  G4  below  A.  Then,  taking  moments  about 
G4,  we  must  have 

Th  =  Pa 

(i) 


DR.   SCHEFFLER'S   METHOD.  805 

and  the  line  of  resistance  can  then  be  drawn  with  this  thrust, 
as  has  been  done  in  the  figure.  Next  take  the  general  case, 
when  the  given  points  are  not  in  these  special  positions.  Let 
them  be  any  two  points,  as  A2  and  Gy 

In  this  case,  the  point  of  application  of  the  thrust  at  the 
crown  is  not  necessarily  Ay  but  may  be  some  other  point  of  the 
crown-joint :  hence  the  quantities  to  be  determined  are  two ; 
viz.,  the  thrust  T  at  the  crown,  and  the  distance  x  of  its  point 
of  application  below  A.  Let  the  combined  weight  of  the  frrst 
two  voussoirs  and  their  load  be  P^  and  the  horizontal  distance 
of  A2  from  a  vertical  line  through  the  centre  of  gravity  of  Pt  be 
at. 

Let  P2  be  the  combined  weight  of  the  first  three  voussoirs 
and  their  load,  and  let  a2  be  the  horizontal  distance  of  G3  from 
a  vertical  line  through  the  centre  of  gravity  of  Pv 

Let  the  vertical  depth  of  A2  below  A  be  klt  and  that  of  G3 
below  A  be  //2.  Then,  taking  moments  about  A2  and  G3  respec- 
tively, we  shall  have 

T(h,  —  x)  =  P,a,     and     T(h2  -  x)  =  P2a2) 

two  equations  to  determine  the  two  unknown  quantities  T  and 
x,  which  can  easily  be  solved  in  any  special  case ;  and  the  result- 
ing line  of  resistance  can  be  drawn,  which  will  pass  through  the 
two  given  points. 

§264.  Scheffler's  Method. — In  using  Scheffler's  method 
of  determining  whether  it  is  possible  to  pass  a  line  of  resistance 
within  a  given  portion  of  the  arch-ring  as  the  middle  third  or 
not,  we  should  proceed  as  follows  ;  viz., — 

First  pass  a  line  of  resistance  through  I,  the  top  of  the 
middle  third  of  the  crown-joint  (Fig.  280),  and  e,  the  inside  of 
the  middle  third  of  the  springing-joint.  If  this  line  lies  wholly 
within  the  middle  third,  it  proves  that  a  line  of  resistance  can 
be  drawn  within  the  middle  third. 

If  this  line  of  resistance  does  not  pass  entirely  within  the 


806  APPLIED  MECHANICS. 


middle  third,  proceed  as  follows :  Suppose  the  line  thus  drawn 
to  be  I  abode,  passing  without  the  middle  third  on  both  sides, 
as  shown  in  the  figure.  Then  from  a,  the  point 
where  it  is  farthest  from  the  extrados  of  the 
middle  third,  draw  a  normal  to  this  extrados,  and 
find  the  point  where  this  normal  cuts  this  extra- 
dos:  in  this  case,  2  is  the  point  in  question.  In 
this  way  determine  also  the  point  7,  where  the 
normal  from  d  cuts  the  intrados  of  the  middle 

FIG.  280.  i'ii  • 

third  ;  then  pass  a  new  line  of  resistance  through 
the  points  2  and  7,  determining  the  thrust  and  its  point  of  ap- 
plication. If  this  new  line  of  resistance  lies  within  the  middle 
third,  then  it  is  plain  that  it  is  possible  to  draw  a  line  of  resist- 
ance within  the  middle  third  ;  if  not,  it  is  not  at  all  probable 
that  it  is  possible  to  draw  such  a  line. 

If  the  line  of  resistance  drawn  through  I  and  e  goes  outside 
the  middle  third  only  beyond  its  extrados,  as  at  a,  we  should 
draw  our  second  line  of  resistance  through  2  and  e  ;  if,  on  the 
other  hand,  it  goes  outside  only  below  the  intrados  of  the 
middle  third,  as  at  d,  we  should  draw  our  second  line  through 
I  and  7. 

In  the  construction,  we  make  use  of  a  slice  of  the  arch  in- 
cluded between  two  vertical  planes  a  unit  of  distance  apart ;  and 
we  take  for  our  unit  of  weight  the  weight  of  one  cubic  unit  of 
the  material  of  the  voussoirs,  so  that  the  number  of  units  of 
area  in  any  portion  of  the  face  of  the  arch  shall  represent  the 
weight  of  that  portion  of  the  arch. 

We  next  draw,  above  the  arch,  a  line  (DD4  in  Fig.  281), 
straight  or  curved,  such  that  the  area  included  between  any 
portion  of  it,  as  D^D2,  the  two  verticals  at  the  ends  of  that  por- 
tion, and  the  extrados  of  the  arch-ring,  shall  represent  by  its 
area  the  load  upon  the  portion  of  the  arch  immediately  below 
it.  This  line  will  limit  the  load  itself  whenever  this  is  of  the 
same  material  as  the  voussoirs  ;  otherwise  it  will  not.  We  shall 
always  call  it,  however,  the  extrados  of  the  load. 


DR.   SCHEFFLER'S  METHOD. 


807 


The  mode  of  procedure  will  best  be  made  plain  by  the  solu* 
tion  of  examples  ;  and  two  will  be  taken,  in  the  first  of  which 
only  one  trial  is  necessary  to  construct  a  line  of  resistance  that 
shall  lie  wholly  within  the  arch-ring,  and,  in  the  second,  two 
trials  are  necessary. 

EXAMPLE. — The  half-arch  under  consideration  is  shown  in 
Fig.  281,  GG4  being  the  intrados,  AA4  the  extrados  of  the  arch, 


D4   Ds 


D., 


FIG.  281. 


und  DD4  the  extrados  of  the  load.     The  arcs  GG4  and  AA4  are 
concentric  circular  arcs.     The  data  are  as  follows  :  — 

Span  =  2(G4O)  =  6.00  feet, 

Rise  =  GO         =0.50  foot, 

Thickness  of  voussoirs  =  AG  =  A4G4       =0.75  foot, 

Height  of  extrados  of  load  above  A  =  AD         =  0.80  foot. 

The  position  of  the  joints  is  not  assumed  to  be  located.  We 
therefore  draw  through  A  a  horizontal  line  AB,  and  divide  this 
into  lengths  nearly  equal,  unless,  as  is  usual  near  the  springing, 
there  is  special  reason  to  the  contrary.  Thus,  we  make  the  first 
three  lengths  each  equal  to  I  foot,  and  thus  reach  a  vertical 


8o8 


APPLIED  MECHANICS. 


through  G4m,  and  then  the  last  division  has  a  length  of  0.24  foot. 
We  have  thus  divided  the  half-arch  and  its  load  into  four  parts ; 
viz.,  GDDtfv  H.D.D^H^  H2D2D3G4,  and  G4D3D4A4,  the  loads 
on  these  respective  portions  being  represented  by  their  areas 
respectively.  We  assume  the  centre  of  gravity  of  each  load  to 
lie  on  its  middle  vertical ;  and  we  then  proceed  to  determine  the 
numerical  values  of  the  several  loads,  the  distances  of  their 
centres  of  gravity  from  a  vertical  through  the  crown,  also  the 
amount  and  centre  of  gravity  of  the  first  and  second  loads 
together,  then  of  the  first,  second,  and  third,  etc. 

The  work  for  this  purpose  is  arranged  as  follows  :  — 


(1) 

(2) 

(3) 

(*) 

(5) 

(6) 

(7) 

(8) 

(9) 

(10) 

o  .is 

1   1 

Length. 

Height. 

Area. 

Lever 
Arm. 

Moment. 

Partial  Sums. 

Area. 

Moment. 

Lever 
Arm. 

I 

I.OO 

1-57 

1-570 

0.50 

0.785 

I 

1-570 

0.785 

0.500 

2 

I.OO 

1.68 

1.  680 

I.50 

2.520 

1  +  2 

3-250 

3-305 

I.OI7 

3 

I.OO 

1.90 

1.900 

2.50 

4-75° 

1  +  2+3 

5-I50 

8.055 

I-563 

4 

0.24 

1.72 

0.413 

3.12 

1.287 

1+2+3+4 

5-563 

9-342 

1.680 

- 

3-24 

- 

5.563 

- 

9-342 

- 

-     ' 

- 

- 

Column  (i)  shows  the  number  of  the  voussoir. 

"        (2)  gives  the  horizontal  lengths  of   the  several  trape- 

zoids. 

(3)  gives  the  middle  heights  of  the  trapezoids. 
"        (4)  gives  the  areas  of  the  trapezoids,  and  is  obtained  by 
multiplying  together  the  numbers  in  (2)  and  (3). 
(5)  gives  the  distances  from  A  to  the  middle  lines  of 

the  trapezoids. 

"  (6)  gives  the  products  of  (4)  and  (5),  giving  the  moments 
of  the  respective  loads  about  an  axis  through  A 
perpendicular  to  the  plane  of  the  paper. 


DR.   SCHEFFLER'S   METHOD. 


809 


Column  (7)  merely  indicates  the   successive    combinations   of 

voussoirs. 
"        (8)  has  for  its  numbers,  — 

i°.  The  area  representing  the  first  load. 
2°.  The  area  representing  the  first  two  loads. 
3°.  The  area  representing  the  first  three. 
4°.  The  area  representing  the  first  four. 
"        (9)  has  for  its  numbers,  — 

i°.  The  moment  of  the  first  load  about  A. 
2°.  The  moment  of  the  first  and  second  loads 

about  A. 
3°.  The  moment  of  the  first,  second,  and  third 

loads  about  A. 
4°.  The  moment  of  the  first,  second,  third,  and 

fourth  loads  about  A. 

(lO)  is  obtained  by  dividing  column  (9)  by  column  (8)  ; 
the  quotients  being  respectively  the  distance 
from  A  to  the  centres  of  gravity  of  the  first,  of 
the  first  and  second,  of  the  first,  second,  and 
third,  and  of  the  first,  second,  third,  and  fourth 
loads. 

The  calculation  thus  far  is  purely  mathematical,  and  merely 
furnishes  us  with  the  loads  and  their  points  of  application  ;  in 
other  words,  furnishes  us  the  data  with  which  to  begin  our 
calculation  of  the  thrust.  Before  passing  to  this,  it  should  be 
said,  however,  that  we  now  assume  the  joints  to  be  drawn 
through  the  points  Alt  A2,  A3,  and  A4,  and  generally  normal  to 
the  extrados  of  the  arch. 

In  this  proceeding,  we,  of  course,  make  an  error  which  is 
very  small  near  the  crown  and  increases  near  the  springing  of 
the  arch  ;  this  error,  in  the  case  of  voussoir  A^A2G^G^  amounts 
to  the  difference  of  the  two  triangles  A2G2H2  and  A.G^,.  A 
manner  of  making  a  correction  by  moving  the  joint  will  be 
explained  later ;  but  now  we  will  complete  our  example,  as  the 


8 10  APPLIED    MECHANICS. 

errors  are  not  serious  in  this  example.  We  now  pass  a  line  of 
resistance  through.**,  the  upper  point  of  the  middle  third  of 
the  crown-joint,  and  y#4,  the  lowest  point  of  the  middle  third  of 
the  springing.  For  this  purpose  take  moments  about  y#4 ;  and 
we  shall  have,  if  T  =  thrust  at  the  crown, 

0.75?*=  (5.563)  (3-075  "  1.68). 

since  5.563  is  the  whole  weight,  and  3.075  —  1.68  is  its  leverage 
about  04. 
Hence 

0-75^  =  (5.563)  (1-395)  =  776o. 
.-.     T  =  10.35. 

Hence  we  proceed  to  draw  a  line  of  resistance  through  <r, 
assuming,  as  the  horizontal  thrust,  10.35.  To  do  this  we  pro- 
ceed as  follows :  From  a,  the  point  of  intersection  of  ay 
with  the  vertical  through  the  centre  of  gravity  of  the  first 
trapezoid,  we  lay  off  ab  to  scale  equal  to  10.35,  anc^  then  lay 
off  bC  vertically  to  scale  equal  to  1.57,  the  first  load  ;  then  will 
Ca  be  the  resultant  pressure  on  joint  AtGlt  and  its  point  of 
application  will  be  P,  which  gives  us  one  point  in  the  line  of 
resistance.  To  obtain  the  point  /*,,  we  lay  off  aal  =  1.017, 
the  lever  arm  of  the  first  two  loads  ;  then  lay  off  albl  —  10.35, 
the  thrust;  then  lay  off  biC1  equal  to  3.25,  the  weight  of  the 
first  two  loads.  Then  will  C1al  be  the  pressure  on  the  second 
joint ;  and  the  point  Plt  or  its  point  of  application,  is  at  the 
intersection  of  Clal  with  A^GV 

Then  lay  off  aa^  —  1.563,  ajb^  —  10.35,  bf^  =  5.150;  and 
Pt,  the  next  point  of  the  line  of  resistance,  is  the  intersection 
of  Cjt^  with  AZG3.  Then  lay  off  aaa  —  1.680,  azbz  =  10.35, 
b^Cz  —  5.563  ;  and  C3a^  is  the  pressure  on  the  springing,  and 
this  will  intersect  A4G^  at  /?4  unless  some  mistake  has  been 
made  in  the  work.  Then  is  ixPP^P^ft^  the  line  of  resistance 
through  a  and  fit<  and  this  lies  entirely  within  the  middle  third. 


SCHEFFLER'S  MODE   OF  CORRECTING    THE  JOINTS.     8ll 


Henee  we  conclude  that  it  is  possible  to  draw  a  line  of  resist- 
ance within  the  arch-ring  without  having  recourse  to  another 
trial. 

§  265.  Scheffler's  Mode  of  Correcting  the  Joints. — The 

following  is  the  approximate  construction  given 
by  Scheffler  for  correcting  the  joint :  Let  DCG 
be  the  side  of  the  trapezoid,  and  CH  the  uncor- 
rected  joint.  From  b,  the  middle  point  of  677, 
draw  Db ;  then  draw  Gc  parallel  to  bD,  and  ch 
parallel  to  CH.  Then  will  ch  be  the  corrected 
joint. 

Conversely,    having    given    the 
joint  CH,  to  find  the  side  of  the  trapezoid  which 
limits  the  portion  of  the  load  upon  it :  through 
C  draw  DG  vertical ;   join  D  with  b,  the  middle 
point  of    GH ' ;    then  draw   Cg  parallel  to  Db ; 
then,  from  g,  drawing  dg  vertical,  we  thus  have 
the  desired  side  of  the  trapezoid. 
§  266.  Another   Example.  — Another  example  will  now  be 
solved,  which  necessitates  two  trials,  and  where  some  of  the 
joints  have  to  be  corrected.     It  is  practically  one  of  Scheffler's. 
The  dimensions  of  the  arch  are  as  follows  :  — 


Half-span 32.97  feet. 

Rise 24.74  feet. 

Thickness  of  ring 5.15  feet. 

Height  of  load  at  crown   .........     8.24  feet. 

Height  of  load  at  springing 33-5°  feet. 


The  arch  may  be  drawn  by  using,  for  the  intrados,  two 
circular  arcs.  Beginning  at  the  springing,  draw  a  60°  arc  with 
a  radius  of  one-fourth  the  span  ;  then,  with  an  arc  tangent  to 
this  arc,  continue  to  the  crown,  the  proper  rise  having  been 
previously  laid  off.  The  work  for  drawing  a  line  of  resistance 


812 


APPLIED   MECHANICS. 


through  the  top  of  the  middle  third  of  crown-joint  and  the 
inside  of  the  middle  third  of  the  springing  will  be  given  with- 
out comment.  It  is  as  follows : 


(1) 

(2) 

(3) 

(*) 

(5) 

(6) 

(7) 

(8) 

(9) 

(10) 

•S'S 

II 

Length. 

Height. 

Area. 

Lever 
Arm. 

Mo- 
ment. 

Partial  Sums. 

Area. 

Moment. 

Lever 
Arm. 

*  -£ 

i 

8.24 

I4.I 

116.18 

4.12 

479 

I 

116.18 

479 

4.12 

2 

8.24 

16.4 

I35-I4 

12.36 

1670 

1+2 

25I-32 

2149 

8.55 

3 

8.24 

18.3 

IS0-79 

2O.6o 

3106 

I  +  ---  +  3 

4O2.II 

5255 

13.07 

4 

4-13 

22.6 

93-34 

26.79 

2500 

i+.  ..+4 

495-45 

7755 

T5-65 

5 

4-13 

27.1 

111.92 

30.92 

346i 

1+...+S 

607.37 

11216 

18.46 

6 

5-14 

347 

178-36 

35-55 

6341 

i+.  ..+6 

785.73 

J7557 

22.34 

- 

- 

- 

78573 

- 

!7557 

- 

- 

- 

•  i 

28.5  T  =  (785.73)  (34.9  -  22.34), 

28.5  r=  9868.77; 
.-.  T  =  346.27. 

Hence  we  construct  the  line  of  resistance  passing  through  the 
top  of  the  middle  third  of  crown-joint  and  the  inside  of  the 
middle  third  of  the  springing,  using  the  thrust  346.27. 

The  construction  is  shown  in  the  figure,  and  is  entirely 
similar  to  that  previously  used.  The  student  will  readily 
identify  this  first  line  of  resistance,  and  will  see  that  it  goes 
outside  the  middle  third  both  above  and  below,  being  farthest 
above  the  extrados  at  the  first  joint  from  the  crown,  and  farthest 
inside  of  the  intrados  opposite  the  first  joint  from  the  spring- 
ing. Hence  we  proceed  to  pass  a  new  line  of  resistance  through 
the  top  of  the  middle  third  of  the  first  joint  from  the  crown, 
and  the  inside  of  the  middle  third  of  the  first  joint  from  the 
springing. 


SCHEFFLERS  MODE   OF  CORRECTING    THE  JOINTS.     813 

For  this  purpose  we  do  not  need  to  make  out  a  new  table, 
as  it  is  not  necessary  to  insert  any  new  joints.  We  need  only 
two  more  dimensions,  i.e.,  the  vertical  depth  of  each  of  these 


FIG.  284. 

points  below  the  top  of  the  crown  :  these  depths  are  respec- 
tively 2.10  and  16.8. 

Hence  we  proceed  as  follows : 
Let  T  =  thrust  at  the  crown  ; 

x  =  distance  of  its  point  of  application  below  the  top  of 
the  crown-joint. 


8 14  APPLIED    MECHANICS. 

1°.  Take  moments  about  the  upper  one  of  the  two  points, 
and  we  have 

T(2.io  —  x)  —  (116.18)  (8  —  4.12)  =  450.778. 

2°.  Take  moments  about  the  lower  one  of  the  two  points, 
and  we  have 

T(i6.S  -  x)  =  (607.37)  (30.5  -  18.46)  =  7312.734- 

Solving  these  equations,  we  obtain 

T  —  466.8,     x  =  1.134. 

Hence  through  a  point  on  the  crown-joint  at  a  distance  1.134 
below  the  top  of  the  middle  third  of  the  crown-joint  draw 
a  horizontal  line,  this  line  being  the  line  of  action  of  the  thrust. 
Then,  making  the  construction  for  a  new  line  of  resistance  just 
as  before,  only  using  this  new  point  of  application  of  the  thrust, 
and  using  for  thrust  466.8,  we  shall  obtain  a  new  line  of  resist- 
ance, which  passes  through  the  desired  points. 

Another  method  of  drawing  a  line  of  resistance  frequently 
pursued  is  the  following  :  — 

After  determining  the  loads  on  the  successive  voussoirs, 
and  also  the  thrust  for  the  particular  line  of  resistance  which 
we  wish  to  draw,  layoff  these  loads  and  thrust  to  scale  in  their 
proper  order  and  directions,  and  construct  a  force  polygon 
(see  §  126),  then  construct  the  corresponding  frame  (equilib- 
rium polygon),  which  shall  have  its  apices  on  the  vertical  lines 
passing  through  the  centres  of  gravity  of  the  loads  as  drawn  in 
the  figure  of  the  arch.  The  intersections  of  the  lines  of  the 
frame  with  the  joints  give  us  points  of  the  line  of  resistance, 
and  the  line  of  resistance  can  be  drawn  by  joining  them. 

Thus,  applying  the  solution  to  Fig.  281,  we  lay  off 
ab  =  1.570,  be  —  i. 680, 

cd  =  1.900,  de  =  0.413. 

Also,  lay  off 

oa  =  5.87, 

and  draw  the  lines  ob,  oc,  od,  and  oe. 


DKA  W1KG   A    LINE   OF  RESISTANCE. 


815 


8i6 


APPLIED    MECHANICS. 


Then  from  A  draw  Aa  parallel  to  oa,  then  draw  aa^  parallel 
to  ob,  a^  parallel  to  oc,  a^as  parallel  to  od,  and  a^at  parallel  to 
oe,  this  last  prolonged  backwards  of  course  passes  through  G4 ; 
then  will  the  line  of  resistance  be  obtained  by  joining  the  points 
APfffl . 

The  last  figure  on  preceding  page  shows  the  same  method 
applied  to  Fig.  284. 

§  267.  Examples. — Four  more  examples  will  now  be  given 
to  be  worked  out  by  the  student.  The  dimensions  are  approx- 
imately those  given  in  some  of  Scheffler's  examples. 

EXAMPLE  I. — Half-span  =  CD  =  65.16  feet,  rise  =  FD  = 
13.85  feet,  AF=  5.32  feet,  AE  —  6.40  feet.  The  arcs  CF  and 
AG  are  concentric  circular  arcs.  Given  width  of  first  five 
horizontal  divisions  of  line  AB,  counting  from  A,  each  10.66 
feet;  width  of  sixth  division,  11.86  feet;  of  seventh,  2.2  feet. 
Determine  the  possibility  of  drawing  a  line  of  resistance  in  the 
arch-ring. 


FIG.  285. 


EXAMPLE  II.— Half-span  =  63.98  feet,  rise  =  FD  =  31.99 
feet,  AF  —  CG  —  5.32  feet,  AE  —  2.13  feet.  The  intradosand 
extrados  of  this  arch  are  seven  centred  ovals,  both  drawn  from 


EXAMPLES. 


sir 


the  same  centres.  Beginning  at  ine  springing,  an  arc  with  i 
radius  of  21  feet  is  drawn,  subtending  39°;  the  curve  is  con- 
tinued by  a  curve  subtending  24°,  and  having  a  radius  of  35.55 
feet.  From  F  an  arc  subtending  10°  is  drawn  from  a  centre 
on  FD  produced,  and  with  a  radius  of  152  feet;  the  curve  is 
completed  by  an  arc  connecting  the  second  and  last. 


FIG.  2 


Given  horizontal  width  of  each  of  first  six  divisions,  counting 
from  A,  10.66  feet ;  horizontal  width  of  seventh  division,  5.32 
feet.  Determine  the  possibility  of  drawing  a  line  of  resistance 
in  the  arch-rins:. 


FIG.  287. 


EXAMPLE   III.  —  Given  span  =  74.18  feet;   rise   =  45.83 


8i8 


APPLIED   MECHANICS. 


feet ;  radius  of  intrados  =  82.42  feet ;  radius  of  extrados  = 
91.18  feet ;  height  of  load  at  crown  =  8.24  feet ;  width  of  each 
of  five  divisions  nearest  crown  =  8.24  feet;  width  of  sixth 
stone  =  4.13  feet.  Determine  the  possibility  of  drawing  a  line 
of  resistance  within  the  arch-ring. 

EXAMPLE  IV.  —  Given  span  =  37.07  feet ;  thickness  of 
ring  —  AB  =  3.08  feet ;  height  of  load  =  BC  =  82.42  feet. 
Determine  the  possibility  of  drawing  a  line  of  resistance  within 
the  arch-ring. 


FIG.  288. 


§  268.  Criterion  of  Stability.  —  It  has  already  been  stated, 
that,  if  a  line  of  resistance  can  be  drawn  within  the  arch-ring, 
then  the  true  line  of  resistance  will  lie  within  the  arch-ring. 


UNSYMMETRICAL   ARRANGEMENT.  819 

With  those  who,  like  Scheffler,  consider  the  material  of  the 
voussoirs  incompressible,  the  criterion  of  stability  of  an  arch  is, 
that  it  should  be  possible  to  draw  aline  of  resistance  within  the 
arch-ring. 

On  the  other  hand,  Rankine  would  decide  upon  the  stability 
of  an  arch  by  determining  whether  a  line  of  resistance  can  be 
drawn  within  the  middle  third  of  the  arch-ring. 

Other  limits  have  been  adopted  instead  of  the  middle  third. 
In  some  cases  the  only  reason  for  deciding  upon  what  these 
limits  should  be  has  been  custom  or  precedent. 

They  might  also  be  determined  so  that  there  should  be  no 
danger  of  exceeding  the  crushing-strength  of  the  stone. 

It  is  needless  to  say  that  the  first  method  is  incorrect ;  for 
the  material  of  the  voussoirs  is  never  incompressible,  and  an 
arch  where  the  true  line  of  resistance  touches  the  intrados  or 
extrados  could  not  stand,  as  the  stone  would  be  crushed. 

Nevertheless,  no  example  will  be  solved  here,  where 
we  determine  the  possibility  of  drawing  a  line  of  resist- 
ance within  any  other  limits  than  the  middle  third,  as  the 
method  of  procedure  is  entirely  similar  to  what  we  have  done, 
the  computation  of  the  entire  table  being  the  same  in  all  cases, 
the  only  difference  occurring  in  the  computation  of  the  thrust 
and  its  point  of  application,  and  the  consequent  construction  of 
the  line  of  resistance.  The  method  to  be  pursued  is,  as  before, 
by  taking  moments  about  the  points  through  which  it  is  desired 
that  the  line  of  resistance  shall  pass. 

§  269  Unsymmetrical  Arrangement.  —  When  the  arch  is 
unsymmetrical,  either  in  form  or  loading,  the  same  criterion  as 
to  being  able  to  pass  a  line  of  resistance  within  the  middle  thir* 
or  other  limits  of  the  arch-ring  will  serve  to  determine  its  sta 
bility.  The  method  of  procedure  differs,  however,  from  the  factt 
that  whereas  we  have  heretofore  found  it  necessary  to  study 
only  the  half-arch  and  its  load,  and  have  had  the  advantage  of 
knowing,  from  the  symmetry  of  arch  and  load,  that  the  thrust  at 


820  APPLIED   MECHANICS. 

the  crown  is  horizontal,  we  have  not  that  advantage  here,  and 
hence  we  must  study  the  entire  arch,  and  we  must  assume  that 
the  thrust  at  the  crown  may  be  oblique,  and  hence  have  a  verti- 
cal as  well  as  a  horizontal  component. 

In  this  case  it  will  be  necessary  to  have  three  instead  of  two 
points  given,  in  order  to  determine  a  line  of  resistance. 

If  we  assume  (Fig.  289)  a  vertical  joint  at  the  crown,  and  let 
P  =  vertical  component  of  the  thrust  at 
the  crown,  A  =  horizontal  component  of 
the  thrust  at  the  crown,  x  =  distance 
of  point  of  application  of  thrust  at  the 
crown  below  upper  point  of  crown-joint, 
FIG  2g  we  have  thus  three  unknown  quantities, 

and  we  shall  therefore  need  three  equa- 
tions to  determine  them. 

In  this  case,  therefore,  we  must  have  three  points  of  the  line 
of  resistance  given,  in  order  to  determine  it ;  and  a  reasoning 
similar  to  that  pursued  in  §  263  would  show  that  a  line  of  re- 
sistance can  always  be  passed  through  any  three  given  points. 

In  performing  the  work,  we  should  need  to  make  out  a  table 
for  the  part  of  the  arch  on  each  side  of  the  crown-joint,  show- 
ing the  loads,  and  centres  of  gravity  of  the  loads,  on  each  vous- 
scir,  and  on  combinations  of  the  first  two,  first  three,  etc. ;  this 
portion  of  the  work  being  entirely  similar  to  that  done  in  the 
case  of  arches  of  symmetrical  form  and  loading,  only  that  we 
require  a  separate  table  for  the  parts  on  each  side  of  the  crown- 
joint. 

When  these  two  tables  have  been  worked  out,  we  next  pro- 
ceed to  impose  the  conditions  of  equilibrium  by  taking  moments 
about  each  of  the  three  points  given. 

Thus,  suppose  that  (as  is  usually  done  first)  we  pass  a  line 
of  resistance  through  the  top  of  the  middle  third  of  the  crown- 
joint  and  the  inside  of  the  middle  third  of  each  springing- 
joint,  we  then  have  only  two  unknown  quantities  to  determine  ; 


YMME  TR1CA  L    A RRA XGEMENT.  52  I 


viz.  f"  and  Qy  inasmuch  as  ;r  becomes  zero.  Hence  we  take 
moments  about  the  inner  edge  of  the  middle  third  of  each  of 
the  springing-joints. 

In  taking  moments  about  the  inner  edge  of  the  middle 
third  of  the  left-hand  springing-joint,  we  impose  the  conditions 
of  equilibrium  upon  the  forces  acting  on  that  part  of  the  arch  . 
that  lies  to  the  left  of  the  crown-joint.  These  forces  are  :  (i°) 
its  load  and  weight,  which  tend  to  cause  right-handed  rotation  ; 
(2°)  the  horizontal  component  of  the  thrust  exerted  by  the 
right-hand  portion  upon  the  left-hand  portion  ;  (3°)  the  vertical 
component  P  of  the  thrust  exerted  by  the  right-hand  portion 
upon  the  left-hand  portion. 

It  is  necessary  to  adopt  some  convention,  in  regard  to  the 
sign  of  P,  to  avoid  confusion  :  and  it  will  be  called  positive 
when  the  vertical  component  of  the  thrust  exerted  by  the  right- 
hand  portion  on  the  left-hand  portion  is  upwards ;  when  the 
reverse  is  the  case,  it  is  negative. 

We  next  take  moments  about  the  inner  edge  of  the  middle 
third  of  the  right-hand  springing-joint,  and  impose  the  condi- 
tions of  equilibrium  upon  the  forces  acting  upon  the  right-hand 
portion  of  the  arch.  In  doing  this,  we  must  observe  that  we 
have  for  these  forces,  (i°)  the  weight  and  load  which  tend  to 
cause  left-handed  rotation  ;  (2°)  the  horizontal  component  Q 
of  the  thrust  exerted  by  the  left-hand  portion  upon  the  right- 
hand  portion, — this  acts  towards  the  right ;  (3°)  the  vertical 
component  P  of  the  thrust  exerted  by  the  left-hand  portion 
upon  the  right-hand  portion ;  and  this,  when  positive,  acts 
downwards. 

Having  determined  the  values  of  Q  and  P,  we  next  proceed 
to  draw  the  line  of  resistance,  by  the  use  of  either  of  the 
methods  employed,  with  symmetrical  arches,  observing  only 
that  the  thrust,  i.e.,  the  resultant  of  P  and  Q,  is  now  oblique, 
and  that  it  acts  in  opposite  directions  on  the  two  sides  of  the 
crown-joint. 


822 


APPLIED   MECHANICS. 


Having  drawn  this  line  of  resistance,  if  we  find  that  it  passes 
outside  of  the  middle  third,  we  draw  normals  through  the  points 
where  it  is  farthest  from  the  middle  third,  and  thus  obtain 
three  points  through  which  to  draw  a  line  of  resistance :  then, 
taking  moments  about  each  of  these  three  points,  we  deter- 
mine, from  the  three  resulting  equations,  values  of  Q,  P,  and 
x,  and  proceed  to  draw  our  new  line  of  resistance;  and,  if  this 
does  not  pass  entirely  within  the  middle  third,  it  is  not  at  all 
probable  that  a  line  of  resistance  can  be  drawn  within  the 
middle  third.  All  the  above  will  be  made  clearer  by  the  fol- 
lowing example : 

EXAMPLE.  —  Given  an  unsymmetrical  circular  arch,  shown 
in  the  figure,  the  intrados  and  extrados  being  concentric 
circles,  EM  —  4',  HF  =  i'.8$,  radius  of  EHF  —  6,  AH  =  0'.$, 
AK  =  c/.8,  to  determine  the  possibility  of  drawing  a  line  of 


/I/I 


FIG.  290. 


resistance  in  the  arch-ring.  The  tables  following  show  the 
mode  of  dividing  up  the  load,  and  getting  the  centres  of 
gravity,  also  the  mode  of  arranging  the  work  for  this  pur- 
pose. 


UNSYMMETRICAL   ARRANGEMENT. 


823 


LEFT-HAND   PORTION. 


Number  of 
Voussoir. 

Width. 

Height. 

Area. 

Lever 
Arm. 

Moment. 

Partial 
Sums. 

Area. 

Moment. 

Lever 
Arm. 

, 

I.OO 

1.32 

1.32 

0.50 

0.660 

I 

1.32 

0.660 

0.50 

2 

I.OO 

I.48 

I.48 

1.50 

2.22O 

I   +   2 

2.80 

2.880 

1.03 

3 

I.OO 

1.84 

1.84 

2.50 

4.600 

1  +  2  +  3 

4.64 

7480 

1.61 

4 

I.OO 

2.42 

2.42 

3-50 

8.470 

i  +...  +  4 

7.06 

I5-950 

2.26 

5 

o-33 

2.63 

0.87 

4.17 

3.628 

I  +  ...+  5 

7-93 

I9-578 

2.47 

- 

- 

- 

7-93 

- 

I9-578 

- 

- 

- 

- 

RIGHT-HAND   PORTION. 


•s  ,. 

4)  *o 

ll 

i 

2 

Width. 

Height. 

Area. 

Lever 
Arm. 

Moment. 

Partial 
Sums. 

Area. 

Moment. 

Lever 

Arm. 

I.OO 
I.OO 

1.32 
1.48 

1.32 
1.48 

0.50 
I.50 

0.660 
2.220 

I   +   2 

1.32 
2.80 

0.660 
2.880 

0.50 
1.03 

- 

- 

- 

2.80 

- 

2.880 

- 

- 

- 

- 

Now  take  moments  about  the  inner  edge  of  the  middle 
third  of  the  left-hand  springing,  and  we  have 

1-732  —  4-ioP  =  7-93(4-10  —  2.47)  =  12.9259. 

Then  take  moments  about  the  inner  edge  of  the  middle  third 
of  the  right-hand  springing,  and  we  have 

0.472  —  i.goP  =  2.80(1.90  —  1.03)  =  2.4360. 
Solving  these  two  equations  gives  us 

Q  =  6.246. 
P  =  0.263. 


824  APPLIED    MECHANICS. 

If  R  represents  the  resultant  of  P  and  Qy  we  have 


R=  V  P*  +  Q*  =6.251; 

hence  we  proceed,  as  follows,  to  pass  a  line  of  resistance  through 
the  top  of  the  middle  third  of  the  crown-joint  and  the  inner 
edge  of  the  middle  third  of  each  springing : 

Through  the  top  of  the  middle  third  draw  a  horizontal  line.. 
Lay  off  aa  —  6.626  and  ab  =  0.263,  and  draw  ab ;  then  ab  = 
6.635  represents,  in  direction  and  magnitude,  the  thrust  at 
the  crown.  Using  this  thrust  in  the  same  way  as  we  did  the 
horizontal  thrust  in  the  case  of  symmetrical  arches,  we  obtain 
the  line  of  resistance  which  is  farthest  outside  of  the  arch  at 
d\  hence,  drawing  a  normal  to  the  arch  from  d,  we  obtain  c,  the 
upper  edge  of  the  middle  third  of  the  first  joint  from  the 
crown.  Hence  we  proceed  to  pass  a  new  line  of  resistance 
through  B,  c,  and  y. 

To  do  this  we  must  assume  Q,  P,  and  x  all  unknown. 
1°.  Take  moments  about  B,  and  we  have 

(1.73  -x)Q  +  4.iP=  12.9259. 
2°.  Take  moments  about  y,  and  we  have 

(0.47  -  x)Q  —  i.$P=  2.436. 
3°.  Take  moments  about  £,  and  we  have 

(0.078  -  x)Q  +  P  =  (1.32)  (0.45)  =  0.594. 
Solving  these  three  equations,  we  obtain 
Q  =  6.905, 
P  =  0.297, 

x  =  0.035. 

Hence  

R=   V  r  +  Q  =  6.91. 

Hence,  if  we  lay  off  a  distance  0.035  below  a,  we  shall  have 
the  point  on  the  crown-joint  at  which  the  thrust  is  applied  ; 


GENERAL   REMARKS.  82 $ 

and  making  the  same  kind  of  construction  as  we  just  made, 
only  using  this  point  instead  of  A,  and  these  new  values  of  Q 
and  Pt  we  construct  the  second  line  of  resistance.  The  con- 
struction is  omitted  in  order  not  to  confuse  the  figure ;  but  the 
line  of  resistance  is  drawn,  and  the  student  can  easily  make 
the  construction  for  himself.  It  will  be  seen,  that,  in  this 
case,  this  new  line  of  resistance  lies  entirely  within  the  arch- 
ring. 

§  270.  General  Remarks.  —  Whenever  there  are  also  hor- 
izontal external  forces  acting  upon  the  arch,  these  should  be 
taken  into  account  in  imposing  the  conditions  of  equilibrium. 

It  will  be  noticed,  that,  in  the  preceding  discussion,  it  has 
always  been  assumed  that  the  load  upon  any  one  voussoir  is  the 
weight  of  the  material  directly  over  that  voussoir.  This  is  the 
assumption  usually  made  in  computing  bridge  arches  :  and  it 
may  be  nearly  true  when  the  height  of  the  load  above  the  crown 
is  not  great ;  but  even  then  it  is  not  strictly  true,  and  when 
this  depth  becomes  great,  as  would  be  the  case  with  an  arch 
which  supports  the  wall  of  a  building,  it  is  far  from  true,  as  the 
distribution  of  the  load  actually  coming  upon  different  parts  of 
the  arch  must  vary  with,  and  depend  upon,  the  bonding  of  the 
masonry,  and  also  upon  the  co-efficient  of  friction  of  the  mate- 
Hal.  Thus,  in  the  case  of  an  arch  supporting  a  part  of  the  wall 
<»f  a  building,  it  is  probable  that  the  only  part  of  the  load  that 
somes  upon  the  arch  is  a  small  triangular-shaped  piece  directly 
)ver  the  arch-,  and  that  above  this  the  material  of  the  wall  is 
supported  independently  of  the  arch.  This  will  be  plain  when 
we  consider,  that,  were  such  an  arch  removed,  the  wall  would 
remain  standing,  only  a  few  of  the  bricks  near  the  arch  falling 
down  ;  and  though  the  number  of  bricks  that  would  fall  would 
be  greater  while  the  mortar  is  green,  still  even  then  only  a  few 
would  drop  out. 

In  regard  to  these  matters,  we  need  experiments ;  but  thus 
far  we  have  none  that  are  reliable. 


826  APPLIED  MECHANICS. 

•       •» 

Then,  again,  we  have  arches  supporting  a  mass  of  sand  or 
gravel ;  and  then  the  mutual  friction  of  the  particles  on  each 
other  comes  into  play,  and  it  is  not  true  in  this  case  that  the 
load  on  any  voussoir  is  the  weight  of  the  material  directly 
above  that  voussoir.  In  some  cases  this  has  been  accounted 
as  a  mass  of  water  pressing  normally  upon  the  arch,  but  we 
cannot  assert  that  such  a  course  is  correct. 

On  the  other  hand,  there  are  cases  where  we  know  that  an 
arch  is  subjected  to  horizontal  as  well  as  to  vertical  forces,  and 
sometimes  we  cannot  tell  how  great  these  horizontal  forces  are. 
Thus,  the  forms  of  sewers  are  an  arch  for  the  top  and  an 
inverted  arch  for  the  bottom  ;  but  in  this -case  the  sides  of  the 
ditch  in  which  the  sewer  is  laid  when  building  it,  are  capable  of 
furnishing  whatever  horizontal  thrust  is  needed  to  force  the 
line  of  resistance  into  the  arch-ring,  provided  that  a  horizontal 
thrust  is  what  is  needed  to  force  it  in.  Hence  it  is,  that,  were 
the  attempt  made  to  pass  a  line  of  resistance  within  the  arch- 
ring  of  almost  any  successful  sewer,  accounting  the  load  as  the 
weight  of  the  earth  above  it,  the  line  would  almost  invariably 
go  outside ;  but  the  earth  on  the  sides  is  capable  of  furnishing 
the  necessary  horizontal  thrust  to  force  it  inside,  unless  a  care- 
less workman  has  omitted  to  ram  it  tight,  or  unless  some  other 
cause  has  loosened  it  on  the  sides  of  the  sewer. 

If  we  know,  in  any  case,  the  actual  law  of  the  distribution 
of  the  load,  we  can  determine  the  proper  form  for  the  arch  by 
the  methods  of  the  first  part  of  this  chapter,  as  was  done  in 
the  case  of  the  parabola  and  of  the  catenary.  Scheffler's 
method  is,  however,  the  one  almost  always  used  for  determin- 
ing the  stability  of  any  stone  arch  against  overturning  around 
the  joints. 

Should  there  ever  arise  a  case  where  there  was  danger  that 
the  resultant  pressure  on  any  joint  made  an  angle  with  the 
joint  greater  than  the  angle  of  friction,  this  could  be  remedied 
by  merely  changing  the  inclination  of  the  joint. 


GENERAL    THEORY  OF   THE  ELASTIC  ARCH.  82/ 

§  271.  General  Theory  of  the  Elastic  Arch.  —  In  the  case 
of  the  iron  arch,  the  loads  upon  the  arch  are  all  definitely 
known  ;  and  it  is  necessary  to  ascertain  with  certainty  the  stress 
in  all  parts  of  the  structure,  and  to  so  proportion  the  different 
members  as  to  bear  with  safety  their  respective  stresses. 

The  general  discussion  of  the  method  used  in  calculating 
such  arches  will  now  be  given ;  the  method  used  being  practi- 
cally that  followed  by  Dr.  Jacob  J.  Weyrauch,  and  explained 
more  at  length  in  his  "Theorie  der  Elastigen  Bogentrager." 

This  discussion  is  also  necessary  in  order  to  prove  the 
proposition  already  enunciated  in  §  262  ;  viz.,  that  "for  an  arch 
of  constant  cross-section,  that  line  of  resistance  is  approximately 
the  true  one  which  lies  nearest  to  the  axis  of  the  arch-ring  as 
determined  by  the  method  of  least  squares." 

In  this  discussion  the  following  definitions  are  adopted  :  — 

i°.  The  axis  of  the  arch  is  a  plane  curved  line  passing 
through  the  centres  of  gravity  of  all  its  normal  sections. 

2°.  The  plane  of  this  axis  is  called  the  plane  of  the  arch. 

3°.  The  axial  layer  of  the  arch  is  a  cylindrical  surface  per- 
pendicular to  the  plane  of  the  arch,  and  containing  its  axis. 

4°.  A  section  normal  to  the  axis  is  called  a  cross-section. 

5°.  The  length  of  the  axis  between  two  sections  is  called 
the  length  of  arch  between  the  sections. 

The  loads  may  be  single  isolated  loads,  or  they  may  be 
distributed  loads. 

We  shall,  in  this  discussion,  assume  in  the  plane  of  the  arch 
a  pair  of  rectangular  axes,  OX  and  O  Y,  positive  to  the  right  and 
upwards  respectively. 

We  will,  then,  assuming  any  point  on  the  axis  of  the  arch 
before  the  loads  are  applied,  call  x,  y,  the  co-ordinates  of  that 
point,  s  the  length  of  axis  from  some  arbitrary  fixed  point,  <£  the 
angle  made  by  the  tangent  line  at  that  point  with,  OX,  r  the 
radius  of  curvature  of  the  axis  at  that  point,  x  -f-  dx,  y  +  dy, 
s  -f-  dsy  and  </>  -f-  dfa  the  corresponding  quantities  for  a  point 


828 


APPLIED   MECHANICS. 


very  near  the  first  before  the  load  is  applied  ;  also  we  will  denote 

by  rj  the  perpendicular  distance  of 
any  fibre  from  the  axial  layer,  by  j, 
the  length  of  arc  measured  to  that 
point  where  this  fibre  cuts  the  cross- 
section  through  (x,  y],  and  s^  -f~ 
ds^  the  length  of  arc  measured  on 

this  fibre  to  the  next  cross-section, 

FlG  w-  so  that  ds  will  be  the  distance  apart 

of  the  cross-sections  measured  on  the  axis,  and  ds^  on  the  other 
fibre.  All  this  is  done  before  the 
load  is  applied,  and  is  shown  in 
Fig.  291 ;  while  the  changes  brought 
about  by  the  application  of  the 
loads  combined  with  change  of  tem- 
perature are  denoted  by  ^/'s,  and 
shown  in  Fig.  292.  Thus,  x,  y,  s, 
and  0  become  respectively  x  -|- 
Ax,  y  -\-  Ay,  s  -\-  As,  and  0  -f-  ^/0. 
Now  the  course  we  are  to  fol- 
low in  the  discussion  is,  to  imagine 
a  cross-section  dividing  the  arch  into  two  parts,  and  to  impose 
the  conditions  of  equilibrium  between  the  external  forces  acting 
on  the  part  to  one  side  of  the  section,  and  the  forces  exerted 
by  the  other  part  upon  this  part  at  the  section.  These  latter 
forces  may  be  reduced  to  the  three  following :  — 

1°.  A  normal  thrust   Tx  uniformly  distributed  over  the  sec- 
tion, the  resultant  acting  at  the  centre  of  gravity  of  the  section. 
2°.  A  shearing-force  Sx  at  the  section, 

3°.  A  bending-couple  at  the  section ;  this  comprising  a 
stress  varying  uniformly  from  the  axial  layer,  and  amounting 
to  a  statical, couple,  tension  below,  and  compressions  above,  the 
axial  layer. 

Moreover,  (i)  and  (3)  combined  amount  to  a  uniformly  vary* 


FIG.  292. 


GENERAL    THEORY  OF  THE  ELASTIC  ARCH.  829 

ing  stress,  the  magnitude  of  whose  resultant  is  Tx>  its  point  of 
application  not  being  at  the  centre  of  gravity  of  the  section  ; 
this  sort  of  composition  having  been  already  exhibited  in  the 
case  of  the  short  strut  (§  207). 

Now,  let  r  be  the  radius  of  curvature  of  the  axial  layer  at 
the  section  ;  and  we  have,  from  Fig.  291,  by  similar  sectors, 

d$n  =•  ds  -f-  r)(  —  d<j>)  =  ds  —  t]d$.  (i) 

But 


Now,  if  the  loads  are  applied,  and  the  changes  take  place 
that  are  indicated  in  Fig.  292,  we  shall  have,  by  suitable  sub- 
stitutions in  (i), 

d(s^  4-  A*,)  =  d(s  4-  AJ)  -  ^d(^  +  A<£)  ;  (3) 

and,  combining  this  with  (i)  and  (2),  we  obtain 

dTAj,  _  (d&s         </A<ft\     r  .  , 

ds,  '-~-(l&  "*  ds)7+j 

Now,  the  change  of  length  of  fibre  from  ds^  to  d(sr)  +  AJ,)  is 
due  to  two  causes:  (i)  the  change  of  temperature,  (2)  the  stress 
acting  on  the  fibre  normal  to  the  section. 

Let  €  =  co-efficient  of  expansion  per  degree  temperature. 
T  =  difference  of  temperature,  in  degrees. 
/,,  =  intensity  of  stress  along  the  fibre  at  section. 
E  =  modulus  of  elasticity  of  the  material. 
Then 


_A  =  *****  =  ld^5  -     ^AcA     r  (  . 

E~     ds^    "\ds        *  ds  Jr  +  rf 

Hence,  solving  for  .p^  we  have 

(6) 


ds  ds 


830  APPLIED   MECHANICS. 

this  being  the  expression  for  the  stress  per  square  inch  on  the 
fibre  whose  distance  is  -rj  from  the  axial  layer. 

Hence  we  shall  have,  by  summation,  if  elementary  area  = 
dA, 

rVA<f>      rndA        d^s      rdA 

T,  =  WA  =  4-/S^  -  -2T  V+-,  + 

and  for  the  moment  Mx  we  have,  by  taking  moments  about  the 
neutral  axis  of  the  section  (i.e.,  horizontal  line  through  its  cen- 
tre of  gravity), 


M.  =  W  =  **  -        a  +  **,.  (8) 


Let  ^4  A  =  A,  r2  =  O,  and  observe  that  ^dA  —  o, 

r  +  V) 

since  the  axis  passes  through  the  centre  of  gravity  of  the  sec- 
tion, and  we  have 

fl 


r  -}-  rj  r  +  f)  r 

2 


r  +  rj  r  '    r    r  + 

Making  these  substitutions,  we  have 
Tx 


Mx 

" 


Hence,  solving  for  —  -  and  —  ?,  we  have 
ds  ds 


GENERAL    THEORY  OF   THE  ELASTIC  ARCH. 

Now,  from  Fig.  292,  we  have 

d(x  4-  A*)  =  d(s  4-  AJ)  cos  (<£  -f  A<p), 
d(y  +  Ay)  =  d(s  4-  A^)sin 

but,  if  we  write  cos  A<£  =  I,  and  sin  A<^>  =  A<^>, 

A6^  -  cos  6       A    sin     -  — 

ds  ds 

dy  dx 

sin(u)  -\~  Au>)    =  sin©  4" 

Hence 


as  as 


+  —  ^  + 

as  \  as 

or,  omitting  the  last  terms,  and  integrating, 

A*  =  -/A<^  +  /K&,  (n) 

Ay  =      f^dx  +  /Kajr/  (12) 

% 

and,  integrating  (9)  and  (10), 

A*   =  /Yds,  (13) 

A<£  =  /.Ytfr.  (14) 

In  these  four  equations  we  have 

AA:  =  horizontal  deflection  due  to  the  loads, 
Ay  =  vertical  deflection  due  to  the  loads, 
AJ-  =  change  of  length  of  arc  due  to  the  loads, 
A<£  =  change  of  slope  due  to  the  loads. 

The  three  equations  which  we  shall  have  occasion  to  use  are 
(11),  (12),  and  (14),  and  if  we  make  the  integrations  between 
the  limits  x  and  o,  they  become,  by  changing  their  order, 


(15) 


8$2  APPLIED  MECHANICS. 

f  *Ydx,  (16) 

t/*  =  0 

(17) 


where  J00  is  the  change  of  slope  for  x  =  o. 
If,  now,  we  write 

.,        Mx         Mx          Tx 


we  shall  have 

X=M1~~1  (20) 

K= -/>,  +  «;  (21) 

or  if  we  neglect  the  effect  of  temperature,  we  may  write 

X  =  M,,  (M) 

Y=-Pl.  (23) 

Moreover,  we   may  with  very  little   error  substitute  the 
moment  of  inertia  /  for  O,  in  the  value  of  Ms,  i.e.,  writing  this 

MK         MK          Tx 


§  272.  Manner  of  using  the  Fundamental  Equations  to 
Determine  the  Stresses  in  an  Iron  Arch.  —  In  order  to  be 
able  to  determine  the  stresses  in  all  the  members  of  an  iron 
arch  with  any  given  loading,  we  need  to  determine  the  three 
quantities  TM  5^,  and  Mx  for  each  section. 

Now,  if  we  let  Rx  represent  the  thrust  at  the  section,  we 
shall  have 


DETERMINA  TION  OF  STRESSES  IN  AN  IRON  ARCH.     833 


R*  =  V^2  4-  Sx* ;  (i) 

and,  if  we  let  Hx  and  Vx  represent  the  horizontal  and  vertical 
components  of  Rx  respectively,  we  have  that  we  need  to  deter- 
mine the  three  quantities  Hx,  Vx>  and  Mx  for  each  section. 

Let  us  suppose  the  arch  to  be  subjected  to  vertical  loads 
only,  and  let 

H  —  horizontal  component  of  thrust  at  all  points, 

V  =  vertical  component  of  left-hand  supporting  force, 

VI  —  vertical  component  of  right-hand  supporting  force, 
M  —  bending-moment  at  left-hand  support, 

M'  =  bending-moment  at  right-hand  support. 

Assume  origin  of  co-ordinates  at  left-hand  support,  and 
x  +  to  the  right,  and  y  -f-  upwards,  and  impose  the  conditions 
of  equilibrium  upon  the  forces  acting  on  the  part  of  the  arch 
between  the  section  and  the  left-hand  support ;  then  we  have, 
if  J^is  any  one  load,  and  a  the  x  of  its  point  of  application, 

H*  =  H,  (2) 

Vx  =   V-  UW,  (3) 

Mx  =  M  +   Vx  -  Hy  -  ^QxW(x  -  a).  (4) 

Hence  it  is  plain  that  the  three  quantities  which  we  need 
to  determine  are  Ht  V,  and  M. 

Now  these  are  also  the  three  unknown  quantities  which  will, 
by  suitable  reductions,  become  the  three  unknown  constant 
quantities  in  equations  (n)  to  (14).  The  determination  of  these 
three  quantities  requires  three  conditions ;  what  these  condi- 
tions are  depends  upon  the  manner  of  building  the  arch,  as  will 
be  seen  from  the  following  three  special  cases  :  — 

CASE  I.  —  Let  the  arch  be  jointed  at  three  points,  viz.,  the 
two  supports,  and  one  other  point  whose  co-ordinates  are  x  = 
x^  and  y  =  yv  Then  we  know,  that,  for  all  points  where 
there  is  a  hinge,  there  can  be  no  bending-moment.  Hence 


834  APPLIED  MECHANICS. 


M  =  o,        M'  =  o,       and      MXi  =  o, 

which  are  the  three  required  conditions  ;  and,  if  these  be  im- 
posed, it  is  easy  to  obtain  Hm  Vx,  and  Mx,  for  every  section. 

CASE  II. — Let  the  arch  be  jointed  only  at  the  ends.  Then 
M  =  M'  =  o  gives  us  two  conditions :  and  for  the  third  we 
have  Al  =  o  ;  i.e.,  if  we  put  /  for  x  in  equation  (16),  §  271, 
after  having  made  the  integrations,  we  have  the  third  equa- 
tion, as  this  expresses  simply  the  condition  that  the  sup- 
ports remain  at  the  same  horizontal  distance  apart  after  the 
load  is  put  on  as  before.  With  these  three  conditions  we  can 
determine  HM  VM  and  Mx  for  all  sections. 

CASE  III. — Let  the  arch  be  fixed  in  direction  at  the  ends* 
We  must  now  have  three  conditions.  These  will  be  as  follows: — 

1°.  Al  =  o  ;  i.e.,  the  supports  remain  at  the  same  horizontal 
distance  apart  after  the  load  is  applied  as  before. 

2°.  Ah  =  o  (h  being  the  difference  of  level  of  the  sup- 
ports) ;  i.e.,  the  supports  remain  at  the  same  vertical  distance 
apart  after  as  before  the  load  is  applied. 

3°.  J0t  =  o;  i.e.,  the  tangents  at  the  ends  make  the  same 
angle  with  each  other  after  as  before  the  load  is  applied. 

The  value  of  A(f>t  is  obtained  by  integrating  (15),  §  271,  and 
then  substituting  /  for  x,  or  h  for  y,  observing  that  A(f>0  =  o. 

The  value  of  Al  is  obtained  by  integrating  (16),  §  271,  and 
then  substituting  /  for  x. 

The  value  of  Ah  is  obtained  by  integrating  (17),  §  271,  and 
then  substituting  /  for  x,  or  h  for  y. 

In  this  case,  if  we  neglect  the  effect  of  temperature,  write 

/  for  /2,  omit  all  terms  containing  -,    and  also  neglect   Tx  in 

(15),  (16),  and  (17)  of  §  271,  we  shall  obtain  by  making  one 
integration, 


DETERMINATION  OF  STRESSES  IN  AN  IRON  ARCH.    835 


/M 
~ 


While  it  has  often  been  proposed  to  use  these  as  approxi- 
mately true,  nevertheless  the  degree  of  approximation  is  too 
coarse  to  render  them  suitable  to  use  in  practice. 

CIRCULAR    ARCH,    UNIFORM    SECTION,    AND    VERTICAL    LOADS. 

We  will  next  deduce  expressions  for  J0,  Ax,  and  Ay,  for  a 
circular  arch  of  constant  cross-section  and  loaded  vertically, 
and  thence  deduce  the  equations  from  which  to  determine  the 
three  quantities  M,  M ' ,  and  H  in  any  such  case,  and  also  the 
expression  for  the  horizontal  thrust  in  an  arch  hinged  at  the 
two  springing-points,  and  symmetrical  in  form.  We  will  write 
in  place  of  £1  the  moment  of  inertia  /,  and  will  neglect  terms 

containing  — ,  in   equations   (15),  (16),  and   (17),  but   will   not 

neglect  Tx. 

Take  the  origin  at  the  left-hand  springing-point,  and  the 
axis  of  x  horizontal. 

Observe  that  if  0  represent  the  angle  the  tangent  line  to 
the  arch  at  the  point  (x,  y]  makes  with  the  axis  of  x,  it  also 
represents  the  angle  subtended  by  the  radius  drawn  through 
the  point  (x,  y)  with  the  vertical  radius,  i.e.,  that  through  the 
crown. 

Let  00  be  the  value  of  0  at  the  origin,  and  let  a  be  the 
value  of  0  at  the  point  of  application  of  any  concentrated  load 
W,  the  co-ordinates  of  this  point  being  (a,  b). 


636  APPLIED  MECHANICS. 

Let  the  co-ordinates  of  the  centre  of  the  circle  be 

g  =  r  sin  00,  —  k  —  —  r  cos  00; 

of  the  crown  be        g  =  r  sin  00,  f  =  r  —  r  cos  00; 
of  the  point  of  ap- 

plication of  W  be  a  —  r(sin  00  —  sin  ar),  b  =  r(cos  a  —  cos  00); 
and    of    any   point 

on  arch,  x  —  /-(sin  00  —  sin  0),  7  =  r(cos  0  —  cos  00). 

The   following  is   a  list   of    relations  which  can  be  easily 
proved,  and  which  are  needed  for  use  in  the  work  that  follows 
them. 
g  —  x  =  r  sin  0;  Tx  =  Vx  sin  0  +  H  cos  0; 

k  -}-  y  •=  r  cos  0;  a>r         ,  -.  ,    Tr         ^        4-  r-r//  \ 

.  J/a;  =  Jf  4-  F-r  —  Zft/  —  2  W(x  —  a)  ; 

x  —  a  =  f(sm  OL  —  sin  0);  o 


y  —  fr  =  r(cos  0  —  cos  <*)  ;  ° 

^  =  -  rd(f>\  M'  =  M+  VI  -  He  -  2  W(l  -  a)\ 

dx  =  —  r  cos  0^/0;  ° 

dy  =  —  r  sin  0</0; 
where  M'  —  bending-moment  at  right-hand  springing-point. 

Also 

X-M*.  Y-er        T* 

X~  - 


By  making  the  substitutions  indicated,  and  also  the  inte- 
grations, we  obtain  from  (15),  (16),  and  (17)  the  following:  — 


=  A(f>0  +  -      {  (00  -  0)(Jf+  Vr  sin  00  +  Hr  cos   00) 

AT 

—  F>-  (cos  0  —  cos  00  )  —  ./fr  (sin  00  —  sin  0)  -f  -2"  JFr  (cos  0 

o 

—  cos  a)  —  2Wr  (a  —  0)  sin  a},  (8) 

o 

=  erx  -  -^^  1  r(sin2  0°  ~  sin2  *)  +  &\s™  0o  cos  00 

X 

—  sin  0  cos  0  +  (00  —  0)]  —^W  (sina  «  —  sin2  0)  [  ~- 
sin  00  +  -»r  cos  00)  [(00  -  0)  cos  0 


DETERMINATION   OF  STRESSES  IN  AN  IRON  ARCH.    83? 

—  (sin  00  —  sin  0)]  --  (cos  0  —  cos00)'  --  [  —  (0o—  0) 
-j-  cos  0  (sin  00  —  sin  0)  +  sin  00  (cos  0  —  cos  00)J 

x  x 

-f-  %r2  W  (cos  0  —  cos  «)'  —  r  cos  02  JF  (a  —  0)  sin  or 

0  O 

-|-  r2  W  sin  a  (sin  «  —  sin  0)  }  ;  (9) 

Ay  =  ery  —  -^  {ZT(sin7  00  -  sin2  0)  -  V  [sin  00  cos  00 

X 

—  sin  0  cos  0  —  (00  —  0)]  —  2  W  [sin  0  cos  0  —  sin  a  cos  a 


4-  #r  cos  00)  [(cos  0  -  cos  00)  —  (00  —  0)  sin  0] 

Vr 

—  \Hr  (sin  00  —  sin  0)a  ---  [—  cos  00  (sin  00  —  sin  0) 

X 

—  sin  0  (cos  0  —  cos  00)  +  (00  —  0)]  -f-  sin  0.2"  ZfV  (a 

o 
x 

—  0)  sin  OL  —  2  Wr  sin  a  (cos  0  —  cos  a) 

o 

—  £2  ZfV  [cos  a  (sin  a  —  sin  0)  +  sin  0  (cos  0  —  cos  a) 

(10) 


We  will  next  write  out  the  same  values  as  applied  to  the 
right-hand  springing-joint  of  a  symmetrical  arch. 

In  this  case  we  have  the  value  of  0  for  the  right-hand  end, 
or  0,  equal  to  —  00;  and  if  we  make  this  substitution,  observing 
that  x  becomes  /  and  y  becomes  zero,  and  if  we  substitute  for 
V  the  value 


V=,^-_M 

r  sin  00  r  sin  00 

then  we  obtain  the  following  :  — 

=  J00  -j-  —  { (Mf  -f  M)  00  —  2Hr  (sin  00  —  00  cos  00) 

—  \^Wr\*a  sin  a—  200  sin  00  -f-  2  (cos  or  —  cos  00)(;  (n) 


838  APPLIED  MECHANICS. 


=  erl- 


+  ~  {  (M'  +  M)  (2  sin  00  -  200  cos  00)  -  Hr(4<l>0  cos1  00 

/ 
—  6  sin  00  cos  00  -}-  200)  +  ^"^V  [20?  cos  00  sin  « 

o 

+  2  cos  00  cos  a  —  200  sin  00  cos  00  +  sin3  00  —  sin*  a 
-2cos'00]j;  (12) 

Ac  =  /^00  +  ~!  (M9  +  M)  (200  sin  00)  -  4^-  (sin8  00 


-  00  sin  00  cos  00)  -  (Jf  -  J/0  (cos  00  - 


200  ("T—  g  --  2  sin*  00  )  +  2a  (2  sin  00  si 
—  4  sin  00  cos  00  -}-  2  sin  or  cos  00  —  2  sin  <*  cos  or 
+4  cos  «  sin  *,]  !  -  ^  {  (*  -  ^')  (cos  00  -  ^ 


2 


sin  ^y  —  2<y—  2  sin  ^  (cosa  —  cos  00)      [.   (13) 

in  0°  _j  ) 


SPECIAL    CASES    OF    SYMMETRICAL    ARCHES. 

1°.  Three-hinged  Arch.  —  In  this  case  we  do  not  need  these 
equations  to  find  the  horizontal  thrust  :  the  proper  ones  can 
be  used  subsequently  if  we  wish  the  deflections  or  slopes. 

2°.  Arch  hinged  at  the  two  springing-points.  —  In  this  case 
M  =  M'  —  o  ;  and  by  making  these  substitutions  in  (12)  and 
solving  for  //,  we  obtain 


cos  00  sin  tf-j-2  cos  00  cos  or—  200  sin  00  cos  0 
-f-  sin8  00  —  sin8  a  —  2  cos8  00] 


0 

0 


-  -t  2  ^(sin8  00  -  sin8  oL)--  (Al  -  erf) 

-  —  -(14) 

400cosa00—  6  sin00cos00+200+—  ^  (200+2  sin00cos00) 


DETERMINATION  OF  STRESSES  IN  AN  IRON  ARCH.    839 

This  formula  gives  the  thrust  when  the  value  of  Al  is  known, 
i.e.,  the  amount  of  relative  yielding  of  the  supporting  points. 

If  the  abutme-nts  do  not  yield  at  all,  then  Al  =  o,  and  that 
term  should  be  omitted  from  the  numerator  ;  so,  also,  if  we 
neglect  a  consideration  of  the  temperature,  then  erl  vanishes 
in  addition. 

Formula  (14)  gives  the  thrust  for  a  set  of  concentrated  loads, 
each  equal  to  W. 

For  a  distributed  load,  we  should  substitute  for  W,  wdx, 
and  integrate  between  the  proper  limits,  w  being  the  intensity 
of  the  load  per  unit  of  horizontal  length,  and  being  constant  or 
variable  according  to  the  distribution  of  the  load. 

The  formula  for  the  thrust  in  the  case  where  the  load  is 
uniformly  distributed  horizontally  (i.e.,  when  w  is  a  constant) 
and  when  it  covers  the  entire  arch  will  now  be  given,  but  will 
not  be  worked  out  here,  as  it  is  easily  obtained  from  (14). 

In  this  formula  the  letters  have  the  same    meanings   as 

heretofore,  and  we  use  also  Al  =  I  ydx  =  area  of  segment  of 

«/o 

arch  ;  and  let  m  =  —,. 

r\.T 

The  formula  is  as  follows  :  — 


-  -f^  +  k\ 

yar     o 


sn  »-       r  cos 


rr  _  ya     /  i  _  __  ^    dc\ 

'  (2r'00  +  JU) 


In  a  similar  way  formulae  are  easily  obtained  for  the  thrust 
when  half  or  a  quarter,  or  some  other  portion  of  the  arch,  is 
loaded. 

3°.  Arch  with  no  hinges.  —  In  this  case,  if  we  know  J0,  Al, 
and  Ac,  or  if  these  are  zero,  we  can  obtain  from  (i  i),  (12),  and 
(13)  the  three  quantities  M,  M',  and  H,  and  then  the  solution  of 
the  arch  follows. 

This  will  not  be  done  here,  however,  as  the  arches  usually 
built  are  of  the  other  kinds. 


840  APPLIED   MECHANICS. 

EXAMPLES. 

i.  Given  a  semicircular  arch  jointed  at  each  springing- joint  and  at 
the  crown,  radius  r.  Trace  out  the  effect  of  a  single  load  W  acting 
upon  it  at  the  extremity  of  a  radius  making  45°  with  the  horizontal. 

Solution. 

The  presence  of  three  joints  gives  us  the  bending-moments  at  each 
of  these  joints  equal  to  zero,  the  co-ordinates  of  these  joints  being 
respectively  (o,  o),  (r,  r),  and  (2^,  o). 

Hence,  using  equation  (4),  we  obtain 

i°.    M  =  o, 

2°.     Vr-  Hr  -   ^(0.7071  ir)  =  o, 
3°.     V(2r)  -   ^(1.707117-)  =  o. 

Solving,  we  have,  therefore, 

V  =  0.85355  W  =  left-hand  supporting-force, 
and 

H  =  0.14645  W  =  horizontal  component  of  thrust. 

Hence   V^  =  0.14645  £F  =  right-hand  supporting-force. 
Hence,  for  a  section  whose  co-ordinates  are  (x,  y), 

x  <  0.29289?-,          Vx  =       0.85355  W; 
x  >  0.29289?-,         Vx  =  —0.14645^. 

Hence  equation  (i)  gives,  for 


x  <  0.292897-,        Rx  =  JFV(o.85355)2  +  (0.14645)* 

=  0.86602  W> 


x  >  0.292897-,         Rx  =  JFV(o.i4645)2  -f  (o.i4645)2 

=  0.2071  1 

Now,  the  angle  made  by  Rx  with  the  horizontal  is,  for 
*  <  0.29289,",  «.  =  "«>-' 


0.29189,1          .,  =  tan--    =  45° 


TRUE  LINE   OF  RESISTANCE   IN  A   STONE  ARCH.       84! 

Knowing,  now,  the  angle  made  by  Rx  with  the  horizontal,  we  can 
find,  for  the  point  (x,  y),  the  angle  made  by  a  tangent  to  the  circle  with 

the  horizon,  or  c^  =  tan-'f Y  Then  resolve  Rx  into  two  com- 
ponents, respectively  tangent  to  the  arch  and  normal  to  it  at  the  point 
x,  y,  and  the  tangential  component  is  the  direct  thrust  Tx,  while  the 
normal  is  the  shearing- force  Sx. 

Then,  for  the  bending  moment,  we  have,  from  (4), 

x  <  0.292897-,    Mx  =  0.85355  Wx  —  0.14645  Wy  ; 

x>  0.29289^-,   Mx  =  0-85355  Wx  —  0.14645  Wy  —  W(x  —  0.29289^). 

Hence  we  determine  the  direct  thrust,  the  shearing- force,  and  the 
bending-moment  at  any  section,  and  can  hence  obtain  the  stresses  at  all 
points. 

2.  Given  the  same  arch  with  a  load  W  distributed  uniformly  over 
the  circular  arc,  find  stresses  at  all  points. 

3.. Given  the  same  arch  jointed  only  at  the  two  springing-points, 
find  stresses  at  all  points. 

§  273.  Position  of  True  Line  of  Resistance  in  a  Stone 
Arch. — The  proof  will  now  be  given  of  the  proposition  already 
referred  to  in  regard  to  the  position  of  the  true  line  of  resist- 
ance ;  viz.,  — 

"  For  an  arch  of  constant  section,  that  line  of  resistance  is 
approximately  the  true  one  which  lies  nearest  to  the  axis  of  the 
arch-ring,  as  determined  by  the  method  of  least  squares." 

PROOF.  —  If  we  denote  by  y  the  ordinate  of  the  axis  of  the 
arch  for  an  abscissa  x,  and  by  /x  that  of  the  line  of  resistance 
for  the  same  abscissa,  then  /*  —  y  is  the  vertical  distance  be- 
tween the  two  curves  for  abscissa  x.  Now,  the  condition  that 
the  line  of  resistance  should  be  as  near  the  arch-ring  as  possi- 
ble, is,  that  the  sum  of  the  (/x  —  y)*  shall  be  a  minimum,  or 

/(/*  — y)**5  =  minimum.  (i) 

But  (#,  /A)  are  the  co-ordinates  of  the  point  of  application  of  the 


842  APPLIED  MECHANICS. 

actual  thrust,  and  hence  (/A  —  y)  is  the  distance  of  the  point  at 
which  the  resultant  thrust  acts  from  the  centre  of  gravity  of 
the  section.  Hence  we  have 


Hence  (i)  becomes 

=  minimum.  (2) 


I  (-TT 


But  H  is  constant  for  the  same  line  of  resistance,  though  it 
varies  for  different  lines  :  hence  we  can  place  H  outside  of  the 
integral  sign.  Hence  we  may  write 

u  =  —  -  I  Mx2ds  ==  minimum.  (3) 

H*j 

Now,  from  (4),  §  272,  we  have 

Mx  =  M  +  Vx  -  Hy  —  ^QxW(x  -  a)  =  <$>(M,  V,  ff); 

M,  V,  and  H  being  constants  for  the  same  line  of  resistance, 
but  varying  for  different  lines.  Hence,  by  differentiating  (3), 
we  have 


du          du  dM,         2    f,,  ,  (• 

JM=JM*-Mf  =  &JM*'lS  '•    J 

du        du   dMr 


=  o,      (4) 
=  o     (5) 


du         du    dM* 


1  =:  -2H-*  C 


dff       dM*   dH 


But  the  first  term  must  be  very  small  :  hence  we  may  write  ap- 

proximately, 

fMxyds  a  o.  (6) 

Now,  the  three  expressions  (4),  (5),  and  (6)  are  identical  with 
(5),  (6),  and  (7)  of  §  272  ;  and  the  conditions  that  these  shall 
be  zero  are,  with  the  degree  of  approximation  there  stated,  the 


DOMES. 


§43 


conditions  that  hold  in  the  case  of  an  arch  fixed  in  direction  at 
the  ends.  Hence  it  follows  that  the  condition  that  the  line  of 
resistance  shall  fall  as  near  the  centre  of  the  arch  as  possible  is 
the  condition  which,  in  an  elastic  arch  fixed  in  direction  at  the 
ends,  gives  us  its  true  position.  Hence  it  would  seem  that 
the  most  probable  position  for  the  true  line  of  resistance  is  the 
nearest  possible  to  the  axis  of  the  arch. 

This  is  the  conclusion  reached  by  Winkler ;  and  a  more 
detailed  discussion  of  the  matter  is  to  be  found  in  an  article 
by  Professor  Swain  in  "Van  Nostrand's  "  for  October,  1880. 

§  274.  Domes.  —  The  method  to  be  used  for  determining 
the  stability  of  a  dome  differs  essentially  from  that  used  in  the 
case  of  an  arch,  for  there  is  no  thrust  at  the  crown  in  a  dome. 
Indeed,  the  most  general  case  is  that  of  the  dome  open  at  the 
top .  we  will,  therefore,  consider  this  case  first  in  studying  the 
action  of  the  forces  required  to  preserve  equilibrium. 

Fig.  293  shows  a  meridional  section  of  an  open  dome, 
pose  that  this  dome  had  been 
entirely  built,  except  the  up- 
per ring-course  of  stones,  rep- 
resented by  LKGH.  Then, 
suppose  that  one  of  the  stones 
only  of  this  course  were  placed 
in  position  without  any  auxil- 
iary support,  its  own  weight 
would  evidently  overturn  it, 
since  the  line  ad,  along  which 
the  weight  acts,  does  not  cut 
the  joint ;  but,  if  the  whole 


Sup- 


ring-course  is  put  in  place,  the 

stones  keep  each  other  in  po-M- 

sition.    The  way  in  which  this 

is  accomplished  is  as  follows : 

they  press  laterally  against  each  other;  and  the  resultant  of  the 


FIG.  293. 


844  APPLIED  MECHANICS. 

pressures  exerted  upon  the  two  lateral  faces  of  any  one  stone 
by  the  other  stones  of  the  course  is  a  horizontal  radial  force, 
which,  combined  with  the  weight  of  the  stone,  gives,  as  the 
resultant  of  the  two,  a  force  which  cuts  the  joint  between  G 
and  H.  Moreover,  sufficient  pressure  will  be  developed  to 
accomplish  this  result,  as  a  failure  to  reach  the  result  will  only 
increase,  the  pressure  upon  the  lateral  faces. 

Moreover,  if,  .when  sufficient  pressure  has  been  developed 
to  bring  the  resultant  of  the  weight  of  the  stone  and  the  above- 
described  horizontal  radial  force  within  the  joint,  it  should 
make  an  angle  with  the  normal  to  the  joint  greater  than  the 
angle  of  friction,  the  tendency  of  the  stone  to  slide  will  increase 
the  lateral  pressure,  and  this  in  turn  will  increase  the  outward 
horizontal  force  till  the  angle  made  by  the  resultant  with  the 
normal  to  the  joint  is  no  greater  than  the  angle  of  friction  of 
the  material  of  the  voussoirs. 

This  will  be  made  plain  by  reference  to  the  figure  (Fig. 
293), where  ab  represents  the  weight  of  the  stone  HLKG,  and 
where  Oft  is  perpendicular  to  HG  and  Oy  is  drawn  so  that  yGfi 
—  <j>,  the  angle  of  friction.  Now,  since  ab  produced  passes  out- 
side of  HG,  horizontal  thrust  must  be  developed.  And,  more- 
over, were  only  sufficient  horizontal  thrust  furnished  to  make 
the  resultant  cut  HG  at  G,  the  angle  between  this  resultant 
and  the  normal  to  the  joint  would  be  greater  than<£;  there- 
fore we  proceed  as  follows  :  assuming  the  horizontal  thrust  to 
act  through  Ly  the  upper  edge  of  the  stone,  we  lay  off  from  b, 
the  intersection  of  the  horizontal  through  L  with  a  vertical  line 
drawn  through  the  centre  of  gravity  of  the  stone,  the  weight  ab 
to  scale,  then  from  b  draw  be  parallel  to  Oy,  and  draw  through 
a  a  horizontal  line  to  meet  be.  Then  will  ac  be  the  horizontal 
force  that  will  be  furnished  by  the  other  stones  of  the  course 
to  keep  this  stone  in  place ;  and  the  pressure  upon  joint  HG 
is  be,  and  acts  at  the  intersection  of  be  and  HG. 

Now   prolong  'be  to  meet  the  vertical  drawn  through  the 


DOMES.  845 


centre  of  gravity  of  the  next  stone,  HGFE,  at  d.  Combine  it 
with  the  weight  of  this  stone ;  this  is  done  by  laying  off  de  —  be, 
and  from  e  drawing  ef  vertical,  and  equal  to  the  weight  of 
FHGE.  The  resultant  fd  makes  an  angle  with  the  normal  to 
FE  greater  than  <£ :  hence  draw  O8  perpendicular  to  FE  and 
Of,  so  that  c0S  =  <j> ;  then  f rom  g,  the  intersection  of  d^"  with  a 
horizontal  line  through  H,  the  top  of  FHGE,  lay  off  gs  =  df9 
through  g  draw  gh  parallel  to  <9e,  and  through  s  draw  sh  hori- 
zontal. Then  is  sh  the  horizontal  thrust  that  will  be  furnished* 
at  H  to  keep  the  stone  HGEF  in  place ;  and  hg  is  the  pres- 
sure upon  joint  FE,  and  acts  at  the  intersection  of  FE  with  hg. 

Next,  prolong  hg  to  meet  the  vertical  through  the  centre  of 
gravity  of  stone  FEDC  at  k ;  lay  off  kl  =  gh,  and  from  /  lay 
off  Im  =  weight  of  stone  FEDC ;  draw  km,  which  cuts  the 
joint  within  the  joint  itself,  and  needs  no  horizontal  thrust  to 
bring  it  inside  ;  hence  mk  is  the  pressure  on  joint  DC. 

Then  draw  mk  to  meet  the  vertical  through  the  centre  of 
gravity  of  ABCD  at  n,  and  lay  off  no  =  km  ;  draw  op  =  weight 
of  ABCD,  and  draw  /;/,  which  will  be  the  pressure  on  the  joint 
BA. 

It  is  necessary,  for  stability,  that  all  these  forces  should  cut 
the  joint  inside  of  the  joint  if  the  stones  are  reckoned  incom- 
pressible ;  or  we  may  adopt  the  middle  third,  or  other  limits,  as 
our  criterion  of  stability. 

As  long  as  it  is  outward  thrust  that  is  required  to  produce 
stability,  it  is  possible  to  furnish  it ;  but,  if  we  should  reach  a 
joint  where  inward  thrust  would  be  required,  this  could  not  be 
furnished,  and  the  dome  would  be  unstable.  Moreover,  the 
resultant  pressure  on  the  springing  gives  us  the  pressure  ex- 
erted upon  the  support  of  the  dome ;  and  it  must  not  cut  any 
joint  of  the  support  outside  of  that  joint,  as  otherwise  the  sup- 
port would  not  stand. 

In  determining  the  numerical  value  and  direction  of  this 
pressure  on  the  support,  we  may  either  construct  it  graphically,. 


846  APPLIED   MECHANICS. 

or  we  may  compute  it  as  follows  :  (i°)  Compound  all  the  ver- 
tical forces,  i.e.,  the  weights,  and  find  the  magnitude  and  line 
of  action  of  the  resultant  of  these.  (2°)  Compound  all  the 
horizontal  forces,  and  find  the  magnitude  and  line  of  action  of 
their  resultant  (in  this  case  the  horizontal  forces  are  two ;  viz., 
ac  applied  at  Z,  and  sk  applied  at  H] ;  then  compound  these  two 
resultants.  The  graphical  and  analytical  method  should  check 
if  no  mistake  has  been  made  in  the  work. 

In  the  above  calculation,  it  has  been  assumed  that  the  figure 
represents  the  portion  of  a  dome  included  between  two  merid- 
ional planes. 

If  we  desire  to  ascertain  the  pressure  exerted  upon  th^ 
lateral  face  of  the  stone  by  its  neighbors  in  the  same  ring- 
course,  we  only  need  to  know  the  angle  made  by  the  two 
meridional  planes  containing  the  lateral  faces  of  the  stone  in 
question,  then  resolve  the  horizontal  thrust  upon  that  stone 
into  two  equal  components,  which  make  with  each  other  an 
angle  equal  to  the  supplement  of  the  angle  of  the  planes ;  i.e., 
resolve  the  outward  horizontal  thrust  into  two  components 
normal  to  the  lateral  faces. 

In  regard  to  the  assumption  that  the  outward  thrust  acts  at 
the  top  of  the  stone,  it  should  be  said  that  this  is  SchefHer's 
custom,  his  reason  being  that  less  thrust  will  be  required  if  he 
assumes  it  at  the  top  than  if  he  assumes  it  nearer  the  middle. 
The  true  position  of  this  thrust  is  probably  much  nearer  the 
middle  of  the  stone. 

An  example  will  next  be  solved,  giving  SchefHer's  method 
of  working. 

EXAMPLE.  —  Given  the  dome  shown  in  the  figure,  sur- 
mounted by  a  lantern  at  the  top ;  determine  whether  it  is 
stable,  and  what  should  be  the  thickness  of  the  support  in  order 
that  the  resultant  pressure  may  not  pass  outside  any  joint  of 
the  pier. 

The  dimensions  are  as  follows  :  — 


DOMES. 


847 


Diameter  of  outer  vertical  circle  =  20  feet. 

Diameter  of  inner  vertical  circle  =  18  feet. 

Angle   made   by  springing-radius   with  vertical  =  75* 
angle  A  OB. 

The  inner  edge  of  the  upper 
voussoir  subtends  18°  on  the 
lower  circle ;  the  width  of  the 
load  of  the  lantern  is  0.6 ; 
the  voussoirs  below  that,  each 
subtend  18°. 

Assume  36  stones  in  a  hori- 
zontal course.  The  width  of  the 
lowest  will,  then,  be  1.51  ;  the 
width  of  the  others  are  deter- 
mined from  their  lever  arms. 

Given  height-  of  pier  =  8 
feet. 

Height  of  the  centre  of  the 
sphere  above  base  of  pier  =  8' 
—  10  sin  15°  =  5.41'. 

The  figure  may  be  taken  to 
represent  the  portion,  of  the 
dome  included  between  two  ver- 
tical planes  passing  through  the 
axis  of  the  dome  :  hence  it 
shows  one  vertical  series  of 
stones. 

We  first  construct  a  table 

giving  the  weights  of  the  differ-  FIG.  294. 

ent  voussoirs  with  any  superin- 
cumbent load,  their  centres  of  gravity,  and  the  moments  of 
their  weights  about   an  axis   passing  through  O,  and  perpen- 
dicular to  the  central  plane  of  the  portion  shown  ;  and  we 
so    choose    our   unit   of    weight    that    the   volumes    of    the 


848 


APPLIED  MECHANICS. 


voussoirs  shall  represent  their  weights.     The  work  is  arranged 
as  follows  :  — 


ELEMENTARY  FORCES. 

HORIZONTAL  FORCES. 

(1) 

(2) 

(3) 

(*) 

(5) 

(6) 

(7) 

(8) 

(9) 

°   1 

Area  of 
Lateral  Face. 

Thick- 
ness. 

Product. 

Lever 
Arms. 

Moment. 

Hori- 
zontal 
Forces. 

Lever 

Arms. 

Moment. 

I 

2 

3 
4 

0.6X6.680 
2.985 

2.985 
2.985 

o-53 
0.74 
1.23 

2.124 
2.209 
3.672 

4-5°7 

3-07 

4-75 
7-05 
8.68 

6.521 
10.493 
25.888 
39.121 

1.74 
1.26 
1.32 

9.60 

9-33 

7.78 

I6.I04 
11.756 
10.273 

- 

- 

- 

12.512 

- 

82.023 

4.32 

- 

38.730 

Column  (i)  contains  the  numbers  of  the  voussoirs,  counting 
from  the  top. 

Column  (2)  contains  the  areas  of  the  lateral  faces  of  the 
stones  shown  in  the  figure.  For  the  three  lower  stones,  the 
area  of  a  ring  subtending  18°  at  the  centre,  and  of  the  dimen- 
sions given,  is  calculated.  For  the  first,  the  height  is  6.68  and 
the  width  0.6. 

Column  (3)  contains  the  thicknesses  of  the  voussoirs  ;  i.e., 
the  length  of  arc  between  their  two  lateral  faces  measured  on  a 
horizontal  circle  through  the  centre  of  gravity  of  the  voussoir, 
which  is  here  taken  at  the  middle  point  of  the  arc  subtended 
by  this  voussoir  on  its  middle  vertical  circle,  i.e.,  one  which 
has  a  radius  9.5  feet. 

Hence,  the  thickness  of  the  lower  stone  being  1.51  feet,  that 
of  the  others  will  be 


DOMES.  849 


Column  (4)  gives  the  weights  of  the  voussoirs  and  their 
loads:  it  is  obtained  by  multiplying  together  the  numbers  in 
columns  (2)  and  (3). 

Column  (5)  gives  the  distances  of  the  centres  of  gravity  of 
the  different  voussoirs  from  the  axis  of  the  dome :  it  may  be 
determined  graphically  or  by  calculation. 

Column  (6)  gives  the  moments  of  the  weights  about  a  hori- 
zontal axis  through  O  perpendicular  to  the  central  plane  of  this 
series  of  voussoirs.  The  graphical  construction  for  determin- 
ing the  horizontal  thrusts  required  is  next  made,  and  the  results 
are  recorded  in  column  (7).  It  will  be  seen  that  no  thrust  is 
required  on  voussoir  No.  4. 

Column  (8)  contains  the  lever  arms  of  these  forces  about 
the  same  axis. 

Column  (9)  contains  their  moments  about  the  same  axis. 

The  construction  thus  far  has  shown  no  case  where  horizon- 
tal tension  instead  of  horizontal  thrust  is  required  to  cause  the 
thrust  on  any  joint  to  pass  within  the  joint :  hence  thus  far  the 
dome  is  stable ;  and  the  question  comes  next  as  to  what  should 
be  the  width  of  the  pier  in  order  that  the  line  of  resistance,  if 
continued  down,  may  remain  within  it. 

For  this  purpose  we  proceed  as  follows  :  — 

Let  /  =  thickness  required. 

Let  breadth  be  equal  to  that  of  the  lowest  voussoir. 

Height  =  8  feet. 

Take  moments  about  the  outer  edge  of  the  base  of  the  pier. 

We  shall  then  have,  — 

i°.  Moment  of  vertical  load  on  dome,  and -of  weight  of  dome 
sector  about  inner  edge  of  springing,  = 

(12.512)  (8. 69  —  6.56)  =  26.52. 
2°.  Moment  of  same  about  outer  edge  of  springing  of  pier  =s 

26.52    +    (I 


850  APPLIED   MECHANICS. 

3°.  Moment  of  horizontal  forces  about  the  same  axis 

38-73°  +  (4-32)  (54i)  =  62.101. 
4°.  Moment  of  weight  of  pier  about  outer  edge  = 

/  =  6.04/2. 


Hence  we  have 

6.04/2  +  12.51/4-  26.52  =  62.101 

/.     /2  +  2.oy/  =  5.89  /.     /  =  i.  60  feet. 

This  is  the  thickness  required  in  order  that  the  line  ot 
resistance  may  remain  within  the  lower  joint. 

If,  on  the  other  hand,  while  pursuing  the  same  method  with 
the  dome  itself,  we  require  that  the  line  of  resistance  shall 
remain  within  the  middle  third  of  the  pier,  we  take  moments 
about  a  point  in  the  springing  of  the  pier  at  a  distance  \t  from 
its  inner  edge,  we  should  then  have 

f/2  +  |(2.o7)/=  5.89 
/.     /2  +  2.07/  =  8.84  .-.     /  =  2.10  feet. 

On  the  other  hand,  we  could  proceed  in  a  similar  way  to 
the  above,  if  we  desired  to  keep  the  line  of  resistance  in  the 
dome  within  the  middle  third,  by  merely  assuming  the  horizon- 
tal thrusts  to  act  at  two-thirds  the  thickness  of  a  joint  from  the 
lower  edge,  and  using  a  point  two-thirds  the  thickness  from 
the  top,  instead  of  the  lower  edge,  as  the  lower  limiting-point 
for  the  pressure  to  pass  through. 

This  will  not  be  done  here,  however. 

EXAMPLE.  —  As  an  example,  St.  Peter's  dome  will  be  given, 
with  the  dimensions  as  given  by  Scheffler  reduced  to  English 
measures.  The  dome  consists  in  its  upper  part,  as  will  be 
evident  from  the  figure,  of  two  domes  ;  the  lantern  resting  on 


DOMES. 


851 


the  two  is  assumed  to  have  one-third  of  its  weight  resting  on 
the  upper,  and  two-thirds  on  the 
lower  dome. 

Diameter  of  dome  =  diameter 
at  the  base  =  144  feet. 

Up  to  a  point  28.48  feet  above 
the  point  Cit  is  formed  of  a  single 
dome  11.84  feet  thick.  In  its 
upper  part,  on  the  other  hand, 
it  is  composed  of  two  domes 
whose  normal  distance  apart  is 
5.15  feet;  the  exterior  having  a 
thickness  of  2.56  feet,  and  the 
inner  of  4.13  feet  at  the  top  and 
5.15  feet  at  the  springing.  At 
the  top  of  these  two  domes  is  an 
opening  12.24  ^eet  radius,  sur- 
mounted by  a  cylindrical  lantern. 
The  magnitude  of  the  load  of  the 
lantern  on  the  dome  is  repre- 
sented on  the  figure  by  1.82  feet 
width  and  56.66  feet  height. 

Height  of  the  entablature 
ABCD  =  23.69. 

Width  of  ABCD  normal  to 
plane  of  paper  =  1.02  feet. 

Thickness  of  ABCD  =  10.30 
feet. 

Divide  the  exterior  dome  into 
nine  parts,  the  interior  into  eight 
of  a  uniform  circumferential  width 


FIG.  295. 


of   10.08  feet,  except  the 
first,  which  has  a  width  of  only  1.82  feet. 

Determine  whether  this  thickness  of  ABCD  is  sufficient  to 
keep  the  line  of  resistance  within  joint  AB. 


852  APPLIED  MECHANICS. 


CHAPTER   X. 
THEORY  OF  ELASTICITY,  AND  APPLICATIONS. 

§  275.  Strains.  —  When  a  body  is  subjected  to  the  action 
of  external  forces,  and  in  consequence  of  this  undergoes  a 
change  of  form,  it  will  be  found  that  lines  drawn  within  the 
body  are  changed,  by  the  action  of  these  external  forces,  in 
length,  in  direction,  or  in  both  ;  and  the  entire  change  of  form 
of  the  body  may  be  correctly  described  by  describing  a  suffi- 
cient number  of  these  changes. 

If  we  join  two  points,  A  and  B,  of  a  body  before  the  exter- 
nal forces  are  applied,  and  find,  that,  after  the  application  of 
the  external  forces,  the  line  joining  the  same  two  points  of  the 
body  has  undergone  a  change  of  length  &(AB),  then  is  the  limit 

of  the  ratio  ,  as  AB  approaches  zero,  called  the  strain  of 


the  body  at  the  point  A  in  the  direction  AB. 

If  AB  -f-  ±(AB)  >  AB,  the  strain  is  one  of  tension  ;  whereas, 
if  AB  -f-  &(AB)  <  AB,  the  strain  is  one  of  compression. 

In  order  to  stut^y  the  changes  of  form  of  the  body,  let  us 
assume  a  point  O  within  the  body  when  there  are  no  external 
forces  acting,  and  let  us  draw  through  this  point  three  rectangu- 
lar axes,  OX,  OY,  and  OZ,  and  assume  a  small  rectangular 
parallelopipedical  particle  whose  three  edges  are  OA,  OBt  and 


STKAIJVS. 


853 


j  and  let  us  examine  the  form  of  this  particle  after  the 
loads  are  applied ;  it  will  be 
found  that  the  edges  OA,  OB, 
and  OC  will  be  of  different 
lengths  from  what  they  were 
before,  and  that  the  angles 
AOB,  AOC,  and  BOCvti\\  no 
longer  be  right  angles,  but 
will  differ  slightly  from  90°. 
Let  the  parallelepiped  oabc- 
g  def  represent  the  form  and 
dimensions  of  the  particle 
after  the  external  forces  are 
applied.  Then  we  shall  have, 

if  e^,  €p  and  ez  represent  the  strains  in  the  directions  OX,  O  Yt 
and  OZ  respectively,  that 

..     .       ,  proj.  og  on  OX  —  OA         ~  A  , 

ex  =  limit  of  -  ~~n~A —  "  as  ^^  approaches  zero, 

OA 

,  pro].  <?£•  on  OY —  OB 
ey  =  limit  of-  -  as  OB  approaches  zero, 


FIG.  296. 


e,  =  limit  of  !£°L 


OZ  —  OC 


C/  0 


as  OC  approaches  zero. 


In  the  figure,  *x  and  ez  are  tensile  strains,  and  iy  is  a  com« 
pressive  strain. 

But  these  strains  do  not  represent  completely  the  distortion 
of  the  particle ;  for  the  plane  CEGD  has  slid  by  the  plane 
OABF  through  the  distance  oc»  the  distance  apart  of  these 
planes  being  OC,  and  the  plane  halfway  between  the  two  has  slid 
just  half  as  far,  so  that  the  amount  of  shearing,  or  the  shearing- 
strain  of  planes  parallel  to  XO  Y  in  the  direction  OX,  may  be 

represented  by  — l-  =  — -  nearly,  or  the  distortion  divided  by  the 
c/C       cCi 


854  APPLIED   MECHANICS. 

distance  apart  of  these  planes.     This,  moreover,  is  the  tangent 
of   the   angle  occlt  or  the  tangent  of   the  angle  by  which  aoc 
differs  from  a  right  angle. 
If,  now,  we  let 
yzx  =  shearing-strain  in  a  plane  perpendicular  to  OZ  in  the 

direction  OX, 
yzy  =  shearing-strain  in  a  plane  perpendicular  to  OZ  in  the 

direction  OY, 
yyx  =  shearing-strain  in  a  plane  perpendicular  to  O  Y  in  the 

direction  OX, 
yyz  =  shearing-strain  in  a  plane  perpendicular  to  0  Y  in  the 

direction  OZ, 
yxz  =  shearing-strain  in  a  plane  perpendicular  to  OX  in  the 

direction  OZ, 
yxy  =  shearing-strain  in  a  plane  perpendicular  to  OX  in  the 

direction  OY, 

and  let  boc  = <j>,  aoc  •=. ft  aob  = x>  then  we  shall 

222 

have 

yzx  =  — '  =  tan  ft  yyz  =  tan  4, 

ccl 

yzy  =  tan  0,  yxz  =  tan  ft 

7^=tanx,  7^=tanx. 

We  thus  have 

yzy  —  yyz  =  tan<^>, 

yxz  =  yzx  =  tan  ft 

7^  =  7^  =  tan  x, 

three  very  important  equations. 

We  thus  have  to  determine  six  strains,  in  order  to  define 
completely  the  state  of  strain  in  a  body  at  a  given  point ;  viz., 
if  we  assume  three  rectangular  axes,  we  must  know  e^,  fy,  cz, 
yzy  =  yyz,  7**-  =  7*z,  y*y  ==  ?,»  three  normal  and  three  tangen- 
tial strains. 


STRAINS  IN    TERMS   OF  DISTORTIONS. 


855 


§  276.  Strains  in  Terms  of  Distortions. — For  the  sake  of 
clearness,  we  will  consider  first  only  the  strains  that  are  parallel 
to  the  z  plane ;  hence,  will  use  only  two  co-ordinate  axes,  OX 
and  OY,  as  shown  in  Fig.  297.  In  this  case  let  us  assume  a 
small  rectangular  particle,  acbd,  the  co-ordinates  of  one  corner 
of  which  are  x,  y,  and  of  the  other,  x  -f-  dx,  y  -f-  dy ;  this  being 


FIG.  297. 

the  case  before  the  load  is  applied.  Let  the  effect  of  the  load 
be  to  move  the  point  a  to  e,  and  b  to  f,  transforming  the 
rectangle  acbd  into  ekfl,  and  thus  changing  x,  y  respectively 
into  x  +  £,  y  +  77,  and  changing  x  -\-  dx,  y  -f-  dy  respectively 
into  x  -\-  g  +  dx  +  ^,  7  +  rf  +  ^  +  afy.  Then  are  dx,  dy 
the  sides  of  the  particle  before  the  load  is  applied.  Then 
from  what  has  preceded  we  shall  have 


drj 


The  first  two  are, evident  at  once.     To  prove  the  third,  ob- 


856 


APPLIED    MECHANICS. 


serve  that  the  shearing-strain,  yxy  is  the  tangent  of  the  angle 
by  which  the  angle  kel  differs  from  a  right  angle ;  hence  it  is 
the  tangent  of  the  sum  of  the  angles  kem  and  len.  Now,  since 
these  angles  are  small,  we  may  take  the  sum  of  the  tangents 
as  nearly  equal  to  the  tangent  of  the  sum.  But 


km      drf 

tan  kem  —  —  =  —  nearly,  and 
em       dx          J 


In      dZ    .      . 
tan  len  =  —  =  —  nearly. 
en       dy          J 


Hence 


FIG.  298. 


In  the  general  case,  Fig.  298,  a  rectangular  parallelepiped i- 


STRAINS  IN   TERMS  OF  DISTORTIONS.  857 

cal  particle,  the  co-ordinates  of  one  corner  of  which  are  x,  y,  z, 
and  of  the  other,  x  +  dx,  y  -f-  dy,  z  +  dz  ;  this  being  the  case 
before  the  load  is  applied. 

Let  the  effect  of  the  load  be  to  change  x,  y,  z,  respec- 
tively, into  x  -f  £,  y  +  17,  z  +  £,  and  to  change  x  -\-  dx,  y  -f-  dy, 
*  +  </*,  into  (*  +  S)  +  (dx  +  dS),(y  +  ri)'+(dy  +  df,\  (z  +  Q 
-f-  (dz  -f  dQ.  Then  are  dx,  dy,  dz,  the  edges  of  the  particle 
before  the  load  is  applied. 

Then,  from  what  has  preceded,  we  shall  have 


d£    ,    ^7  <#       dt  d-n       dt 

^  =  ^  =  _  +  _,  ^  =  T^  =  _  +  ^  yj..?v.  J  +  J. 

The  first  three  will  be  evident  at  once.  As  to  the  last  three, 
the  proof  is  similar  to  that  just  used  in  the  case  of  two  co- 
ordinate axes. 


dy       dx  dz       dx  dz       dy 

§  277.  Determination  of  the  Strain  in  any  Given  Direc- 
tion.—  Suppose  we  are  required,  knowing  the  strains  *x,  ty,  fz, 
7.rp  7*z>  7yz>  to  determine  the  strain  in  a  direction  making  angles 
«>  ft  y>  with  OX,  O  Y,  OZ  respectively.  Assume  our  rectangu- 
lar parallelopipedical  particle  in  such  a  way  that  the  diagonal 
from  (x,  y,  z)  to  (x  +  dx,  y  -\-  dy,  z  +  dz)  shall  be  in  the 
required  direction,  and  call  the  length  of  this  diagonal  ds  (Fig. 
298)  ;  then  we  shall  have 


.  x 

COSa  =  -,  (,) 


858  APPLIED  MECHANICS. 


,  (3) 

dz 
cosy  =  -.  (4) 

Let  e  be  the  strain  in  the  required  direction  ;  then  length  c  £ 
diagonal  after  load  is  applied  will  be 


and  we  shall  have 

!  +  €)2  =  (dx 


or 


(5) 


Now,  subtracting  (i)  from  (5),  and  neglecting  ?(&)*, 
(d<i)2,  and  (d§*  as  being  very  small  compared  with  the  rest,  we 
have 


or 

c^f  =  ^|cosa  +  drjcosfi  +  d£cosy.          (7) 
But 


859 


Hence,  substituting  these,  we  have,  after  dividing  by  ds,  and 
observing  (2),  (3),  and  (4), 


€  =       cos*  a  + 


d-n 


or,  making  use  of  §  276,  we  have 

e  =  €^COS2a  +  ey  COS2  ft  +  «z  COS2  y  -f  yyz  COS  /?  COS  y 

+  y*z  cos  a  cos  y  +  yxy  cos  a  cos  yS,         (12) 

which  gives  us  the  strain  in  any  direction. 

It  can  be  shown  that  there  are  three  directions,  at  right 
angles  to  each  other,  that  give  the  maximum  strains  or  mini- 
mum strains  :  and  we  might  deduce  the  ellipsoid  of  strains,  in 
which  semi-diameters  of  the  ellipsoid  represent  the  strains  ;  but 
we  will  pass  on  to  the  consideration  of  the  stresses. 

§  278.  Stresses.  —  When  a  body  is  subjected  to  the  action 
of  external  forces,  if  we  imagine  a  plane  section  dividing  the 
body  into  two  parts,  the  force  with  which  one  part  of  the  body 
acts  upon  the  other  at  this  plane  is  called  the  stress  on  the 
plane  ;  and,  in  order  to  know  it  completely,  we  must  know  its 
distribution  and  its  direction  at  each  point  of  the  plane.  If  we 
consider  a  small  area  in  this  plane,  including  the  point  O,  and 
represent  the  stress  on  this  area  by/,  whereas  the  area  itself  is 

represented  by  a,  then  will  the  limit  of  —  ,  as  a  approaches  zero, 

be  the  intensity  of  the  stress  on  the  plane  under  consideration 
at  the  point  O.  Observe  that  we  cannot  speak  of  the  stress  at 
a  certain  point  of  a  body  unless  we  refer  it  to  a  certain  plane 
of  action  :  thus,  if  a  body  be  in  a  state  of  strain,  we  do  not 
attempt  to  analyze  all  the  molecular  forces  with  which  any  one 


86o 


APPLIED  MECHANICS. 


particle  is  acted  on  by  its  neighbors  :  but,  when  we  assume  a 
certain  plane  of  section  through  the  point,  the  stress  on  this 
plane  at  the  point  becomes  recognizable  in  magnitude  and 
direction  ;  and  what  the  magnitude  and  direction  of  the  stress 
at  the  given  point  is,  depends  upon  the  direction  of  the  plane 
section  chosen,  the  magnitude  and  direction  differing  for  differ- 
ent plane  sections  through  the  point. 

§279.  Simple  Stress.  —  A  simple  stress  is   merely  a  pull 
or  a  thrust.     Assume  a  prismatic  body,  with  sides  parallel  to 
OX,  subjected  to  a  pull  in  the  direction  of   its 
length  ;  the  magnitude  of  the  pull  being  P.     As- 
sume first  a  plane  section  AA   normal  to   the 
N  direction  of  P,  and  let  area  of  AA  be  A.     Then, 
/    if  px  represent  the  intensity  of   stress   at   any 
point  of  this  plane, 


T\ 


X^ 


_ 

P*  -'• 


This,  which  is  the  intensity  of  the  stress  as  dis- 
tributed over  a  plane  normal  to  its  direction,  may 
be  called  its  normal  intensity. 

On  the  other  hand,  if  we  desire  to  ascertain 

the  intensity  of  the  stress  on  the  oblique  plane  BB,  making  an 

angle  0  with  AA,  we  shall  have 


FIG.  299. 


Area  BB  = 


COS0 


Hence,  if  pr  represent  the  intensity  of  the  stress  on  this  plane 
in  the  direction  OX,  we  shall  have 


pr  = 


COS<9  = 


(I) 


If  we  resolve  this  into  two  components,  acting  respectively  nor 


COMPOUND  STRESS. 


861 


mal  and  tangential  to  BB,  and  if  we  denote  the  normal  intensity 
by/«,  and  the  tangential  by/,,  we  shall  have 

pn   =  pr  COS  0   =  px  COS  2  6,  (2) 

pt  =  pr  sin  0  =  px  cos  0  sin  0.  (3) 

If,  now,  we  assume  another  oblique  plane  section,  perpen- 
dicular to  the  first,  we  shall  obtain  the  normal //and  the  tan- 

gential //  stress  on  this  plane  by  substituting  for  0,  -  —  6 ; 

hence  we  obtain 

A'=Asina0,  (4) 

//  =  /^costfsintf.  (5) 

Hence  follows 

A'  =  A; 

or,  the  tangential  components  of  a  simple  stress  on  a  pair  of 
planes  at  right  angles  to  each  other  are  equal. 

§280.  Compound  Stress.  —  A  compound  stress  may  be 
accounted  to  be  the  resultant  of  a  set  of  simple  stresses,  and 
may  be  analyzed  into  different  groups  of  simple  stresses. 

PROPOSITION.  —  Whatever  be  the  external  forces  applied  to  a 
body,  if  through  any  point  we  pass  three  planes  of  section  at  right 
angles  to  each  other,  the  tangential  components  of  the  stress  on 
any  two  of  these  planes  in  directions  parallel  to  the  third  must 
be  of  equal  intensity. 

To  prove  this  proposition,  assume 
three  rectangular  axes,  origin  at  O,  and 
assume  a  rectangular  parallelopipedical 
particle,  as  shown  in  the  figure,  so 
small  that  we  may  without  appreciable 
error  assume  the  stress  on  any  one  of 
the  faces  to  be  the  same  as  that  on  the 
opposite  face ;  resolve  these  stresses, 
i.e.,  the  forces  exerted  upon  the  faces  of  the  particle  by  the 
other  parts  of  the  body,  into  components  parallel  to  the  axes. 


862  APPLIED   MECHANICS. 

Let  <rx  =  intensity  of  normal  stress  on  the  x  plane, 
a-y  =.  intensity  of  normal  stress  on  the_^  plane, 
<rz  =  intensity  of  normal  stress  on  the  z  plane, 
rKy  =  intensity  of  shearing-stress  on  x  plane  in  direction 

OY, 
rxs  =  intensity  of  shearing-stress  on  x  plane  in  direction 

oz,  ' 

ryx  =  intensity  of  shearing-stress  on  y  plane  in  direction 

ox, 

vyz   =  intensity  of  shearing-stress  on  y  plane  in  direction 

OZ, 
Ttx  =  intensity  of  shearing-stress  on  z  plane  in  direction 

OX, 
Tzy  =  intensity  of  shearing-stress  on  z  plane  in  direction 

OY. 

We  have  thus  apparently  nine  stresses,  which  must  be  given, 
in  order  to  define  the  stress  at  the  point  O  completely  ;  but  we 
will  now  proceed  to  prove  that 


In  the  figure,  the  only  ones  of  these  stresses  that  are  repre- 
sented are  the  following  :  — 


Zy     =  zlyl    —  crz. 

The  other  four  are  omitted,  in  order  not  to  complicate  the 
figure. 

Now,  it  is  evident  that  the  total  normal  force  on  the  face 
AFGD  and  the  normal  force  on  the  face  OBEC  balance  each 
other  independently,  and  likewise  with  the  other  normal  forces. 


GENERAL    REMARKS.  863 


The  only  forces  tending  to  cause  rotation  around  OZ  arc 
the  equal  and  opposite  parallel  forces  rxy  (area  AFGD),  one  act- 
ing on  the  face  AFGD,  and  the  other  on  the  face  OBEC  ;  and 
the  equal  and  opposite  forces  ryx  (area  FBEG),  one  acting  on 
the  face  FBEG,  and  the  other  on  the  face  CO  AD. 

The  first  pair  forms  a  couple  whose  moment  is  rxy  (area 
AFGD)  (xx^  and  the  second  has  the  moment  ryx  (area  FBEG) 

' 


But 

Area  AFGD  =  (FA)  (zz,),     area  FBEG  =  (FB)  (zzt) 

.-.     rxy(FA)  (zzt)  (xx,)  =  ryx(FB)  (**,)  (yy>). 
Cancelling  zz»  we  have 

xXl)  =  ryx(FB)(yyi). 


But 

FA  =  j^x     and 


Q.  E.  D. 

In  a  similar  manner  we  can  prove 


yz 


GENERAL    REMARKS. 

From  what  precedes,  it  follows,  that,  when  we  have  the  six 
stresses 

°"JTJ       °"jfj       °"z)       rxy)       TXZ>       fyZ) 

or,  in  other  words,  the  normal  and  tangential  components  of 
the  stresses  on  three  planes  at  right  angles  to  each  other,  given, 
the  state  of  stress  at  that  point  is  entirely  determined  ;  and, 
when  these  are  given,  it  is  possible  to  determine  the  direction 
and  intensity  of  the  stress  on  any  given  plane. 


864  APPLIED    MECHANICS. 


Moreover,  if  three  rectangular  axes,  OX,  OY,  and  OZ,  be 
assumed,  and  the  direct  strains  along  these  axes  be  given,  and 
also  the  shearing-strain  about  these  axes,  then  the  direct  strain 
in  any  given  direction  can  be  determined,  and  also  the  shearing- 
strain  around  this  direction  as  an  axis. 

The  two  above-stated  propositions  furnish  two  of  the  funda- 
mental propositions  of  the  theory  of  elasticity,  the  third  being 
the  determination  of  the  relation  between  the  stresses  and  the 
strains. 

§  281.  Relations  Governing  the  Variation  of  the  Stresses 
at  Different  Points  of  a  Body.  —  If  we  assume  a  point  whose 
co-ordinates  are  (x,  y,  z),  and  a  small  parallelopipedical  particle 
having  this  point  and  the  point  (x  -{-  dx,  y  -f-  dy,  z  +  dz)  for  the 
extremities  of  its  diagonal,  we  shall  have,  for  the  edges  of  this 
particle,  dx,  dy,  dz,  respectively. 

Now  let  the  stresses  at  (x,  y,  z)  be 


i.e,  <rx  denotes  the  normal  stress  on  a  plane  perpendicular  to 
OX,  and  passing  through  the  point  (x,  y,  z),  etc.  Then,  for 
the  planes  passing  through  (x  -f-  dx,  y  +  dy>  z  +  dz),  we  shall 
have  the  stresses 

o>  +  d<rx,    <ry  -f  d<ry,    <rz  -h  d<jz,    rxy  -f  dTxy,    rxz  +  drxz,    ryz  -f-  dryz. 

We  may  also  have  outside  forces  acting  upon  the  particle  in 
question  :*if  such  is  the  case,  let  the  components  of  the  result- 
ant external  force  along  the  axes  be  respectively 

Xdxdydz,          Ydxdydz,         Zdxdydz. 

Now  impose  the  conditions  of  equilibrium  between  all  the 
forces  acting  on  the  particle.1  To  do  this,  place  equal  to  zero 
the  algebraic  sum  of  all  the  forces  parallel  to  each  of  the  axes 


RELATIONS  BETWEEN  STRESSES  AND  STRAINS.        865 

respectively,  the  moment  equations  having  already  been  incor- 
porated in  our  demonstration  that 

rxy  ==   Tyxi  TXZ   =   rzx>  ryz   —   Tzy. 

Hence  we  have  three  conditions  of  equilibrium,  as  follows  :  — 

(  a  x  +  dax  —  ax)dydz  +  (rxy+  drxy  —  rxy]dxdz  +  (rxz  +  drxz  —  rxz)dxdy  +  Xdxdydz  =  o, 
(oy  +day  —  ay}dxdz  +  (^xy~\-drxy—rxy}dydz  +(ryz  -}-dryz  —  ryz  )dydx+  Ydxdydz  =  o, 
(oz  +daz  —Gz)dxdy  +  (ryz+dTyz  —ryz  )dxdz  +  (rxz+dTXZ  —  rxz)dzdy  +  Zdxdydz  =  O. 

Hence,  reducing,  and  dividing  by  dxdydz,  we  have 

**  +*p  +  z~+x=0i  (1) 

dx  dy          dz 


= 


dx          dy          dz 

%•+*&+*•.  +2=0.  (3) 

dx  dy          dz 

If  the  particle  is  in  the  interior  of  the  body,  and  we  dis- 
regard its  weight,  then  X  =  Y  =  Z  =  o. 

Equations  (i),  (2),  and  (3)  give  the  necessary  relations  which 
the  variations  of  stress  from  point  to  point  must  satisfy  in  order 
that  the  conditions  of  equilibrium  may  be  fulfilled. 

§  282.  Relations  between  the  Stresses  and  Strains.  — 
Before  proceeding  to  the  general  problems  of  composition  of 
stresses,  i.e.,  of  determining  from  a  sufficient  number  of  data 
the  stress  upon  any  plane,  we  will  first  discuss  the  relations 
between  the  stresses  and  the  strains  ;  and  we  will  confine  our- 
selves to  those  bodies  that  are  homogeneous,  and  of  the  same 
elasticity  throughout. 

From  what  we  have  already  seen,  if  to  a  straight  rod  whose 
cross-section  is  A  there  be  applied  a  pull  P  in  the  direction  of 


866  APPLIED   MECHANICS. 

its  length,  the  intensity  of  the  stress  on  the  cross-section  wi 
be 

P 


and,  if  E  be  the  tensile  modulus  of  elasticity  of  the  material  of 
the  rod,  the  strain  in  a  direction  at  right  angles  to  the  cross- 
section,  or,  in  other  words,  in  the  direction  of  the  pull,  will  be 


Now,  another  fact,  which  we  have  thus  far  taken  no  account 
of,  is,  that  although  there  is  no  stress  in  a  direction  at  right 
angles  to  the  pull,  or,  in  other  words,  although  a  section  at 
right  angles  to  the  above-stated  cross-section  will  have  no  stress 
upon  it,  yet  there  will  be  a  strain  in  all  directions  at  right  angles 
to  the  direction  of  the  pull  :  and  this  strain  will  be,  for  any  direc- 
tion at  right  angles  to  the  pull, 


being  of  the  opposite  kind  from  € ;  thus,  if  «  is  extension,  «x  is 
compression,  and  vice  versa. 

Hence,  if,  at  any  point  O  of  such  a  rod,  we  assume  three 
rectangular  axes,  of  which  OX  is  in  the  direction  of  the  pull, 
and  we  use  the  notation  already  adopted,  we  shall  have 

P 

0>     =      — ,        <Ty     =     <TZ     =     rxy     =     rxz     =     Tyz     —     O, 

./i 


Jc,  m  £  m 

y**  =  yy*  =  o. 


RELATIONS  BETWEEN  STRESSES  AND   STRAINS.        867 
MODULUS    OF    SHEARING    ELASTICITY. 

In  the  case  of  direct  tension  or  compression,  when  only  a 
simple  stress  is  applied,  we  have  defined  the  modulus  of  elas- 
ticity as  the  ratio  of  the  stress  to  the  strain  in  its  own  direction. 

Adopting  a  similar  definition  in  the  case  of  shearing,  we 
shall  have 


where  G  is  the  modulus  of  shearing  elasticity. 


GENERAL    RELATIONS    BETWEEN    STRESSES    AND    STRAINS. 

Whenever  a  compound   stress  acts  on  a  body  at  a  given 
point,  let  the  stresses  be 


TXZ> 


then  we  shall  have,  for  the  strains. 


€*=  "F  "    ™  "^  ~  «"*> 

xi         w  -£i         w  /t 


€>=-- Z__^ 

^  77*  C^  Z7 

xi  fH  tL  Wl  JtL 


_  _ 

*"  E~    mE  ~  mE' 


This  enables  us  to  determine  the  strains  in  terms  of  the 
stresses,  as  soon  as  the  values  of  E,  G,  and  m  are  known  from 
experiment,  for  the  material  -under  consideration. 

If,  on  the  other  hand,  the  stresses  be  required  in  terms  of 
the  strains,  we  can  consider  e^,  cy,  cg,  yxy,  yxzy  yyz,  as  known,  and 
determine  o-^,  a-y,  <rg,  rxy,  ryzt  rxg,  from  the  above  equations. 


868  APPLIED   MECHANICS. 

We  thus  obtain 

(i) 


m 


(2) 


m 

~^>  <3) 


and,  by  solving  these  equations  for  the  stresses,  we  have 


orv  = 


m  + 
m 

m  4- 


,,x 
- 

and  also 

T^y  =   Gyxy,      (7)          T^2  =   Gyxz,      (8)          T^  =  ^y^.      (9) 

These  equations  express  the  stresses  in  terms  of  the  strains. 
The  three  last  might  be  written  as  follows  (see  §  276) :  —     ' 


(12) 

as  these  forms  are  often  convenient. 

§  283.  Case  when  a-z  =  o.  —  Inasmuch  as  there  are  many 
cases  in  practice  where  the  stress  is  all  parallel  to  one  plane, 
and  where,  consequently,  the  stress  on  any  plane  parallel  to 
this  plane  has  no  normal  component,  it  will  be  convenient  to 
have  the  reduced  forms  of  equations  (4),  (5),  and  (6)  which 
apply  in  this  case. 


VALUES  OF  E,   G,   AND  m.  869 

Let  the  plane  to  which  the  stresses  are  parallel  be  the  Z 
plane ;  then  o-,  =  o.     Then  equation  (6)  becomes 


m  —  2 

:*+  ev 


m-  i)  ' 

and,  substituting  this  value  of  ez  in  (4)  and  (5),  and  reducing,  we 
obtain 


which  are  the  required  forms. 

The  other  three  equations,  viz.,  — 


ry    =  ^,  «  „,  y, 


remain  the  same  as  before. 

§284.  Values  pf  £,  G,  and  m  --  These  three  constants 
need  to  be  known,  to  use  the  relations  developed  above. 

i°.  As  to  E,  this  is  the  modulus  of  elasticity  for  tension, 
and  has  been  determined  experimentally  for  the  various  mate- 
rials, as  has  been  already  explained.  Moreover,  it  has  also  been 
shown  experimentally,  that,  with  moderate  loads,  the  modulus 
of  elasticity  for  compression  is  nearly  identical  with  that  for 
tension  in  cast-iron,  wrought-iron,  and  steel. 

2°.  As  to  m,  in  those  few  applications  that  Professor  Ran- 
kine  gives  of  his  theory  of  internal  stress,  such  as  the  case  of 
combined  twisting  and  bending,  he  determines  the  greatest  in- 
tensity of  the  stress  acting  ;  and  his  criterion  is,  that  this  shall 
be  kept  within  the  working-strength  of  the  material.  This  is 
equivalent  to  assuming  m  —  oo.  The  more  modern  writers, 


8/0  APPLIED  MECHANICS. 

such  as  Grashof  and  others,  take  account  of  the  fact  that  m  has 
a  finite  value,  and  make  their  criterion  that  the  greatest  strain 
shall  be  kept  within  the  quotient  obtained  by  dividing  the  work- 
ing-strength by  the  modulus  of  elasticity  of  the  material. 

Thus,  if  f  is  the  working-strength,  and  o-x  the  greatest 
stress,  and  e,  the  greatest  strain,  Rankine's  criterion  of  safety 
is 


whereas  the  more  modern  criterion  is 


The  resulting  formulae  differ  in  each  case ;  and,  as  has  been 
stated,  those  of  Rankine  could  be  derived  from  the  more  gen- 
eral ones  by  making 

i 

-  =  o     or     m  —  oo, 

which  is  never  the  case. 

As  to  the  value  of  m,  but  few  experiments  have  been  made. 
Those  of  Wertheim  give,  for  brass,  2.94 ;  for  wrought-iron,  3.64. 

The  values  ^  =  3  and  m  =  4  are  those  most  commonly 
adopted,  so  that 

-L  -=  I  —  =  T 

m       3  m       4* 

3°.  The  value  of  G,  the  shearing-modulus  of  elasticity,  i.e., 
the  ratio  of  the  stress  to  the  strain  for  shearing,  has  been 
determined  experimentally,  and  has  generally  been  found  to  be 
about  two-fifths  that  for  tension. 

According  to  the  theory  of  elasticity,  we  must  have 

f, i       m      & 

as  may  be  proved  as  follows  :  — 


VALUES  OF  E,   G,   AND   m.  871 

Assume  a  square  particle  whose  side  is  ay  and  let  a  simple 
normal  stress  a-  be  applied  at  the  face  AB ;  then  we 
shall  have,  on  the  planes  BD  and  AC,  a  shearing-stress 

(§  279)  A- 

T  =  <r  sin  45  cos  45    =  $<r. 

On  the  other  hand,  if  we  let 


r 

a-  I 

€  =   2?>  FIG-  301- 

the  strain  of  the  particle  in  the  direction  AD  will  be  e,  while 

that  in  the  direction  AB  will  be  — -  ;  hence  the  particle  will 

m 

become  a  rectangle,  the  side  AD  changing  its  length  from  a  to 
a  -f-  ac,  and  side  AB  changing  from  a  to  a . 

The  diagonals  will  no  longer  be  at  right  angles  to  each 
other ;  and,  if  we  denote  by  a  the  angle  by  which  their  angle 
differs  from  a  right  angle,  we  shall  have,  for  the  shearing-strain 
on  the  planes  AC  and  BD, 

y  =  tana. 

But,  after  the  distortion,  the  angle  ADB  will  become 


.%    tan 


therefore,  dividing,  and  carrying  the  division  only  to  terms  of 
the  first  degree,  we  have 


-  2  tan-  =  i  -fi  -\--\ 

2  V  ^/ 

(a\  m  +  I 

;)     -^r(- 


8/2  APPLIED   MECHANICS. 

But 

y  =  tan  a  =  2  tan  -  nearly 


but 


and     - 


,,       i 

•  .     Cr  = 


2  #2   4-    I 

§  285.  Conjugate  Stresses.  —  If  the  stress  on  a  given  plane 
at  a  given  point  of  a  body  be  in  a  given  direction,  the  stress  at 
the  same  point  on  a  plane  parallel  to  that  direction  will  be 
parallel  to  the  given  plane.  Let  YO  Y  represent,  in  section,  a 
given  plane,  and  let  the  stress  on  that  plane  be  in  the  direction 
XOX. 

Consider  a  small  prism  ABCD  within  a  body,  the  sides  of 
whose  base  are  parallel  respectively  to  XOX  and  YOY.  The 
forces  on  the  plane  AB  are  counterbalanced  by  the  forces 
on  the  plane  DC  ;  the  resultants  of  each  of  these  sets  being 
equal  and  opposite,  and  acting  along  a  line  passing  through  O. 
Hence  the  forces  acting  on  the  planes  AD  and  BC  must  be  bal- 
anced entirely  independently  of  any  of  the  forces  on  AB  or 
DC:  and  this  can  be  the  case  only  when  their  direction  is  paral- 
lel to  YOY;  for  otherwise  their  resultants,  though  equal  in 
magnitude  and  opposite  in  direction,  would  not  be  directly 
opposite,  but  would  form  a  couple,  and,  as  there  is  no  equal  and 
opposite  couple  furnished  by  the  forces  on  the  other  faces,  equi- 
librium could  not  exist  under  this  supposition. 

§  286.  Composition  of  Stresses.  —  The  general  problem  of 
the  composition  of  stresses  may  be  stated  as  follows  :  — 


PROBLEM. 


8/3 


Knowing  the  stresses  at  a  given  point  of  a  strained  body 
on  three  planes  passing  through  that  point,  to  find  the  stress  at 
the  same  point  on  any  other  plane,  also  passing  through  the 
same  point.  The  stresses  on  the  three  given  planes  are  not 
entirely  independent ;  in  other  words,  we  could  not  give  the 
stresses  on  these  three  planes,  in  magnitude  and  direction,  at 
random,  and  expect  to  find  the  problem  a  possible  one.  Thus, 
suppose  that  the  planes  are  at  right  angles  to  each  other,  we 
have  already  seen  that  we  have  the  right  to  give  their  three 
normal  components,  o-^,  o-y,  and  a-z,  and  the  three  tangential, 
Txy,  ryzt  and  Txg,  and  that  ryx  =  Txy,  etc.  We  will  now  proceed 
to  special  cases. 

§  287.  Problem.  —  Given  the  three  planes  of  action  of  the 
stress  as  the  x,  y,  and  z  plane  respectively,  and  given  the  nor- 
mal and  tangential  components  of  the  stresses  on  these  planes, 
viz.,  a-*,  a?,  o-g,  rxy,  TXZ,  and  ryz,  to  find  the  intensity  and  direction 
of  the  stress  on  a  plane  whose  normal  makes  with  OX,  OY, 
and  OZ  the  angles  a,  /?,  and  y  respectively,  where,  of  course, 

COS2  a  +  COS2/?  -f-  COS2y  =    I. 

Draw  the  line  ON,  making  angles  a,  /?,  and  y  with  OXt  O  Y, 
and  OZ  respectively  ;  then  draw 
near  O  the  plane  ABC  perpen- 
dicular to  ON.  It  has  the  direc- 
tion of  the  required  plane,  and 
cuts  off  intercepts  OA,  OB,  and 
OC  on  the  axes ;  and,  moreover, 
we  shall  have,  from  trigonometry, 
the  relations, 

Area  BOC  =  (ABC)  cos  a, 

Area  AOC  =  (ABC)  cos/?, 

Area  A  OB  =  (ABC)  cosy. 


FIG.  3°2- 


Now  consider  the  conditions  of  equilibrium  of  the  tetrahe- 
dron OABC.     The  stress  on  ABC  must  be  equal  and  directly 


8/4  APPLIED  MECHANICS. 

opposed  to  the  resultant  of  the  stresses  on  the  three  faces 
AOC,  BOC,  and  AOB.  Now  let  us  proceed  to  find  this  result- 
ant. 

In  the  direction  OX  we  have  the  force 


rxy(AOC) 
=  (ABC)(<Txcosa.  4- 

Lay  off  OD  to  represent  this  quantity.    In  the  same  way  repre- 
sent the  force  in  the  direction  O  Y  by 

OE  =  <ry(AOC)  4-  ryz(BOA)  +  rxy(BOC) 

=  (A£C)(a-j,cosp  4-  T^  cos  a  -|-  7>cosy), 

and  that  in  the  direction  OZ  by 
OF  =  <rz(AOB)  + 


Now  compound  these  three  forces,  and  we  have,  as  resultant 
force, 


R  =  OG  =  V2  4-  OS2  + 
and  as  resultant  intensity 


ABC  ABC 


4-  (ay  cos  ^8  -f  r^cosa  +  r^cosy)* 

-f  (o-zcosy  4-  r^cosa  4-  Tyzcosf!)2l 
=  V^tf^cos2**  4-  a-/cos2^  4-  o-z2cos2y 

4-  rjr/(cos2a  4-  cos2  /?)  4-  T^22(cos2a  4-  cos2y) 

4-  Ty/(cos2/5  4-  cos2y)  4-  2  0-^(7-.^  cos  /3  4-  T^  cos  y)  cos  a 

4-  2<Ty(T^y  cos  a  4-  T^Z  cos  y)  cos  /3 
4-  2o-2(r^z  COS  a  4-  Ty2  COS  /?)  COS  y  4~  sr^r^  COS  (3  COS  y 

4-  2TjcyTyZ  cos  a  cos  y  4-  2TyzTrzcosacos/:?!  ; 


SrXESS£S  PARALLEL    TO  A   PLANE.  875 

the  direction  being  given  by  the  angles,  ar,  (3r,  and  yr,  where 

COSar   =    — ,        COSfir   =    —  ,         COSyr    =    — . 

K  K  K 

§  288.  Stresses  Parallel  to  a  Plane.  — To  solve  the  same 
problem  when  there  is  no  stress  in  the  direction  OZ,  and  when 
the  new  plane  is  perpendicular  to  XOYt  or,  in  other  words,  in 
the  case  when  the  planes  of  action  are  all  perpendicular  to  one 
plane,  to  which  the  stresses  are  all  parallel :  we  then  have 

cr2  =  rxz  =  Tyz  =  O      and      ft  =   90°  --  a, 

and  hence 

o"  =  Vo>2COS2a  +  a-/ sin2  a  +  rxy2  -f   2  (o^  +  o>) rxy  COS  a  sin  a. 

Or  we  may  proceed  as  follows  :  — 

Let  the  normal  intensity  of  the  stress  on  the  x  plane  (i.e., 
that  perpendicular  to  OX)  be  a-xt  that  on  the 
y  plane  <ry,  and  the  tangential  intensity  rxy. 
Let  ON  be  the  direction  of  the  normal  to 
the  plane  on  which  the  stress  is  to  be  deter- 
mined, and  let  the  angle  XON  =  a.  Then 
let  the  plane  AB  be  drawn  perpendicular  to 
ONy  and  let  us  consider  the  equilibrium  of  G 

the  forces  exerted  by  the  other  parts  of  the 
body  upon  the  triangular  prism  whose  base  is  ABO  and  alti- 
tude unity. 

If  we  compound  the  forces  acting  on  the  faces  AO  and  OB, 
we  shall  have,  in  their  resultant,  the  total  force  on  the  face  AB 
in  magnitude  and  direction.  Moreover,  we  have  the  relations, 

Area  OB  =  area  AB  cos  a     and     Area  OA  =  area  AB  sin  a. 
Force  acting  on  OB  in  direction  OX  = 
Force  acting  on  OB  in  direction  OY  = 
Force  acting  on  OA  in  direction  OX  =  rxy(OA)t 
Force  acting  on  OA  in  direction  OY  =  <ry(OA). 


876  APPLIED  MECHANICS. 

Hence,  if  we  lay  off 

OD  =  <rx(OB)  +  rxy(OA)     and     OC  =  <ry(OA)  +  rxy(OB), 

then  will  (9/>  represent  the  total  force  acting  in  the  direction 
OX,  and  <?£r  will  represent  the  total  force  acting  in  the  direc- 
tion OY. 

Compounding  these,  we  shall  have  OE  as  the  resultant  total 

force  on  the  face  AB,  and  —  —  will  represent  its  intensity. 


To  deduce  the  analytical  values,  we  have 


OD  =  <rx(OB)  -f-  rxy(OA)   =  (AB)(<rxCO$a  +  r 

OC  =  <ry(OA)   +  rxy(OB)  =  (AB)  (a-y  sin  a    -f  r^COSa) 


/.    OE  =  \1OD2  +  OC2 


COS  a  -|-  r^sina)2  +  (a-^sina  -f-  r 
=  ABy\<rx2  cos 2  a  -f-  ay2  sin2  a  +  2rxy  cos  a  sin  a(o-jr  +  o>) 

4-  r^./(cos2a  -f-  sin2  a)  \. 

Or,  if  o>  represent  the  resultant  intensity  on  the  plane  AB,  and 
OT  the  angle  this  resultant  makes  with  OX,  we  shall  have 


-f-   2Txy(<rx  -f-  a-y)  COS  a  sin  a  -{-  Txy2\,      (l) 

and 

OC 


Moreover^  it  is  sometimes  desirable  to  resolve  the  stress  into 
normal  and  tangential  components.  If  this  be  done,  and  if  a 
and  r  represent  respectively  the  normal  and  tangential  com- 
ponents, we  shall  have 

OF  EF 


PRINCIPAL   STRESSES. 


but 

OF  =  ODcosa  +  £Dsma     and     EF  —  ££>cosa  —  O£>sina 

OD  OC  . 

/.       o-  =    -  COS  a   -f     --  Sin  a 
AB  AB 

=  o-^COS2a  -h  <Tysin2a  H-  2r^  COS  a  sin  a      (2) 
and 

OC  OD  . 

r  =  -  COS  a   —    --  sma 
AB  AB 

=  (o-j,  —  tr*)  cos  a  sin  a  -f-  r^(cos2a  —  sin2  a) 

(<Ty      -      CTjA 
—  -  J  sin  2a  +  TT,  cos  2a.    (3) 

§  289.  Principal  Stresses.  —  It  will  next  be  shown,  that, 
whatever  be  the  state  of  stress  in  a  body,  provided  the  stresses 
are  all  parallel  to  one  plane,  the  planes  of  action  being  all  taken 
perpendicular  to  this  plane,  there  are  always  two  planes,  at  right 
angles  to  each  other,  on  which  there  is  no  tangential  stress  ; 
these  two  planes  being  called  the  planes  of  principal  stress,  the 
stress  on  one  of  these  planes  being  greater,  and  the  other  less, 
than  that  on  any  other  plane  through  the  same  point. 

To  prove  the  above,  it  will  be  necessary  only  in  the  last 
case,  which  is  a  perfectly  general  one,  to  determine  for  what 
values  of  a  the  value  of  r  is  zer.o,  and  whether  these  values  of  a 
are  always  possible.  We  have 

o>  —  <r*  . 

T  =     -  —  SHI  2d   -J-   T^  COS  2tt  .• 

and,  if  we  put  this  equal  to  zero,  we  have 
sin  2  a  2rx 


COS  2a 


=  tan  2a  = 


and  this  gives  us,  for  all  values  of  o-^,  <ryt  and  rxr  two  possible 
values  for  2a,  differing  from  each  other  by  180°,  hence  two 
values  for  a  differing  by  90°.  Hence  follows  the  first  part  of 
the  proposition. 


8; 8  APPLIED   MECHANICS. 


The  latter  part  — that  these  are  the  planes  of  the  greatest 
and  least  stresses  •—will  be  shown  by  differentiating  the;  value 
of  o>2,  and  putting  the  first  differential  co-efficient  equal  to  zero ; 
and,  as  this  gives  us 

—  2OY*  cos  a  sin  a  -{-  2  ay*  cos  a  sin  a 
+  2.Txy(vx  -f-  (Ty)(cos2a  —  sin2  a) 

=  2  (arx  +  o>)  \  (<?>  —  <TX)  COS  a  sin  a  -f-  r^COS*  a  —  sin2  a)£ 

=  o, 

therefore  we  have  the  same  condition  for  the  maximum  and 
minimum  stresses  as  we  have  for  the  planes  of  no  tangential 
stress. 

It  follows  that  the  determination  of  the  greatest  and  least 
stresses  at  any  one  point  of  a  body  is  identical  with  the  deter- 
mination of  the  principal  stresses ;  and  it  will  be  necessary, 
whenever  the  stresses  on  any  two  planes  are  given,  to  be  able 
to  determine  the  principal  stresses,  as  one  of  these  is  the 
greatest  stress  at  that  point  of  the  body,  and  the  other  the 
least. 

§  290.  Determination  of  Principal  Stresses.  —  When  the 
stress  is  all  parallel  to  one  plane,  viz.,  the  z  plane,  and  when 
the  stresses  on  two  planes  at  right  angles  to  each  other  are 
given,  i.e.,  their  normal  and  tangential  components,  we  may  be 
required  to  determine  the  principal  stresses.  Proceed  as  fol- 
lows :  Given  normal  stresses  on  X  and  Y  planes  respectively, 
o>  and  a-?,  and  tangential  stress  on  each  plane  rx)n  to  find  prin- 
cipal stresses. 

From  §  288  we  have,  for  a  plane  whose  normal  makes  an 
angle  a  with  OX, 

tr*.  =  Vcr/cos2**  H-  (Ty2  sin2  a  +  tTxy(vx  -f  CT^,) cos  a  sin  a  -j-  r^y2,    (i) 
orn  =  <jx  COS2  a  +  (Ty  sin2  a  -|-   2rxy  COS  a  sin  a,  (2) 

T  =  r^(c082a  —  sin2  a)  —  (a-x  —  cr,) COS  a  Sill  a,       (3) 


DETERMINATION  OF  2'RINCITAL   STRESSES.  8/9 

or 

&x  —    <?y    .  f    x 

T   =    Try  COS  20. —Sin  2tt.  (4) 

Now,  the  condition  that  the  plane  shall  be  a  plane  of  prin- 
cipal stress  is,  that  r  —  o.     Hence  write 

Tyy (COS2  a  —  sin2  a)  —  (crr  —  ay)  COS  a  sin  a  =  O, 

find  a,  and  substitute  its  value  in  (2),  and  we  shall  have  the 
principal  stresses.  The  operation  may  be  performed  as  fol- 
lows ;  viz.,  — 

From  (3)  we  have 

vx  —  <?y 

(a)      2  COS2  a  —  I   =  COS  a  sin  a 


xy 


I     (  <TX     —     <Ty                             ,               \ 

.'.      COS2  a  =   -  <  I  H £  COS  a  Sin  a  > 

2(  T^ 

(b)     1—2  sin2  a  =  - 


1  (            (Tx  —  a-y  ) 
.*.     sin2a  =  -<i —  cos  asm  a  >. 

2  I  rxy 


Hence 


crx  -f-  <T>         COS  a  sin  a  j  o-^2  —   2crxvy  -f-  ^jy2 

o-»  =  —       -  H <  - 


2  (  T^ 

or 

&x  +  °"v         COS  a  Sin  a 


( 

(O",   - 


But  we  have,  since  (4)  equals  zero, 

tan  20.  =  - 

(7 x     —     <Ty 

±2T^ 


.%    sm  2a  =  2  sm  a  cos  a 


8 80  APPLIED   MECHANICS. 

Hence  substitute  for  cos  a  sin  a  its  value,  and 

cr x  ~\~  (Ty         I   i 
2  2 

which  gives  us  the  magnitudes  of  the  principal  stresses ;  the 
plus  sign  corresponding  to  the  greater,  and  the  minus  sign  to 
the  less. 

EXAMPLES. 

i.  Let,  in  the  last  section,  a-y  =  o,  and  find  the  principal  stresses. 
Here  we  have 

tan2a  =  — — 
and 

<r    =  —  ±  l-\!7~r~- 


2.  Given  two  principal  stresses,  to  find  the  stress  on  a  plane  whose 
normal  makes  an  angle  a  with  OX. 

In  this  case  rxy  =  o. 

Hence  we  have  the  case  of  §  288,  with  the  reduction  of  making 
rxy  =  o.  We  may  therefore  obtain  the  result  by  substitution  in  the 
results  of  §  288,  or  we  may  proceed  as  follows  :  — 

(a)  Find  stress  on  new  plane  in  direction  OX;  this  will  be,  §  279, 

<rx  cos  a. 
(£)  Find  stress  on  new  plane  in  direction  OY ';  this  will  be,  §  279. 

a-y  sin  a. 
(c)  Compound  the  two,  and  the  resultant  is 

oy  =  vWJCOS2a  +  a-/ sin2  a.  (i) 

{d)  Normal  component  of  <rx  cos  a  is 

<rx  COS2  a. 

(f)  Normal  component  of  <ry  sin  a  is 

cr*  sina  a. 


ELLIPSE   OF  STRESS. 


881 


(/)   Add,  and  we  have,  for  normal  stress, 

a-n  =  a-x  cos2  a  +  o-y  sin2  a. 
(g )  Tangential  component  of  o-^  cos  a  is 
—vx  cos  a  sin  a. 

(Ji)  Tangential  component  of  ay  sin  a  is 
+<ry  cos  a  sin  a. 

(£)  Add,  and  we  have,  for  tangential  stress, 
T  =  (o>  —  or -r)  cos  a  sin  a. 


(3) 


§291.  Ellipse  of  Stress.  — In  the  case  above,  i.e.,  when 
the  two  principal  stresses  are  o>  and  cry  respectively,  if  we 
represent  them  graphically  by  OA  = 
a-*  and  OB  =  o-y,  and  let  CD  be  the 
plane  on  which  the  stress  is  required, 
its  normal  making  with  OX  the  angle 
XON  =:  a,  then,  from  what  has  been 
shown,  if  OR  represent  the  intensity  of 
the  resultant  stress  on  this  plane,  we 
shall  have 


OR  =  oy  =  VW2  COS2  a  -f  a-/  sin2  a  ; 

and,  moreover, 

OE  =  ovcosa,        OF 


<rv  sin  a. 


If  we  denote  these  by  x  and  y  respectively,  letting  (xyy)  be 
the  point  R,  i.e.,  the  extremity  of  the  line  representing  the 
stress  on  AB,  then 


X  = 


(x\2 
-}  =  cos2  a        and 
<W 


y  =  ory  sm  a, 
=  sin*  a 


882  APPLIED   MECHANICS. 

which  is  the  equation  of  an  ellipse  whose  semi-axes  are  vx  and 
<ry  respectively ;  hence  the  stress  on  any  plane  will  be  repre- 
sented by  some  semi-diameter  of  the  ellipse. 


SPECIAL  CASES. 

I.  When  the  two  given  stresses  are  equal,  or  o-*  =  o-y,  then 


(Tr  =  \a-x2  COS2  a  +  o>2  sin2  a  =  crx, 

and 

ox  COS  a 

cos  OT  =  —       -  =  cos  a        and        sincv  =  sin  a; 


therefore  the  stress  is  of  the  same  intensity  on  all  planes,  and 
always  normal  to  the  plane. 

II.  When  the  two  given  stresses  are  equal  in  magnitude 
but  opposite  in  sign,  or  <ry  =  —  <r*,  then 

0>    =    (Tx. 

But 

cos  OT  =  cos  a        and        sinar  =  —sin  a, 
hence 

a.r  —    — a/ 

therefore  the  stress  on  any  plane  whose  normal  makes  an  angle 
a  with  OX  is  of  the  same  intensity  cr^,  but  makes  an  angle 
equal  to  a  with  OX  on  the  side  opposite  to  that  of  the  normal 
to  the  plane. 

PROBLEM.  —  A  pair  of  principal  stresses  being  given,  to 
find  the  positions  of  the  planes  on  which  the  shear  is  greatest. 

Solution.  —  Let  T  =  (<ry  —  <TX)  sin  a  cos  a  =.  max. 

Therefore  differentiate,  and 

cos2  a  —  sin2  a  =  o 

A     COS  a  =    ±sina  /.     a  =  45°  OF  135°. 


SPECIAL   MODES  Cf  SOLUTION  OF  SOME   PROBLEMS.    883 

§  292.  Some  Special  Modes  of  Solution  of  some  Prob- 
lems.—  The  case  where  two  principal  stresses,  o-^  and  <ry,  are 
given,  to  find  the  stress  on  any  plane  whose  normal  makes  an 
angle  a  with  OX,  may  be  solved  as  follows,  graphically :  — 

Let,  Fig.  304,  <rx  —  OA,  and  a-y  =  OB.     Let  XON  =  a. 

Now, 

CT      -f-    <TX  CT      —    <Tjf 


Hence,  instead  of  proceeding  at  once  to  find  the  resultant 
stress  on  CD  due  to  the  action  of  <rx  and  a-y,  we  may  first  find 
that  due  to  the  action  of  the  two  equal  principal  stresses  of  the 
same  kind, 

Q>  4-  cr  x 


then  that  due  to  the  pair 


and         _ 


2  2 

and  then  the  resultant  of  these  two  resultants. 

The  first  resultant  will  be  evidently  laid  off   on    ON,  and 

equal  in  magnitude  to  —  -  —  ;  hence  let  O  M  =  —  --  -,  and 
OM  will  be  the  first  resultant. 

The   second  resultant  will  be   of   magnitude  — 
will  have  a  direction  MR  such  that  the  angle  NMS  —  SMR. 

Hence,  laying  off  this  angle,  and  making  MR  =  —  -  -9 

we  shall  have  for  the  final  resultant,  OR,  as  before. 

This  construction  will  be  useful  in  the  following  case:  — 
To  find  the  most  oblique  stress,  we  must  find  for  what 

value  of  a  the  angle  MOR  is   greatest.     This  will   be   made 


884  APPLIED   MECHANICS, 

evident  if  we  observe,  that,  for  all  positions  of  the  plane,  the 


triangle  OMR  has  always  OM  —  ^L±L-f  and  MR  - 

both  of  constant  length.  Hence,  if,  with  M  as  a  centre  and 
MR  as  a  radius,  a  circle  were  described,  and  a  tangent  were 
drawn  from  O  to  this  circle,  the  point  of  tangency  being  taken 
for  R,  then  will  OR  be  the  most  oblique  stress  ;  i.e.,  the 
stress  is  most  oblique  when  ORM  =  90°.  Therefore  greatest 
obliquity  = 


sin-1 


-y  -h  a-x 


§  293.  Converse  of  the  Ellipse  of  Stress The  converse 

of  the  ellipse  of  stress  would  be  the  following  problem  :  Given 
any  two  planes  passing  through  the  point  in  question  ;  given 
the  intensities  and  directions  of  the  stresses  on  these  planes, 
—  to  find  the  principal  stresses  in  magnitude  and  in  direction. 

The  first  step  to  be  taken  is,  to  assure  ourselves  that  the 
conditions  are  not  incompatible,  as  they  are  liable  to  be  if  the 
planes  and  stresses  are  taken  at  random.  The  test  of  this 
question  is,  to  resolve  each  stress  into  two  components,  respec- 
tively parallel  to  the  two  planes  ;  and,  if  the  conditions  are  not 
inconsistent,  the  components  of  each  stress  along  the  plane  on 
which  it  acts  must  be  equal.  The  proof  of  this  statement  can 
be  made  in  a  similar  way  to  that  used  in  proving  that  the 
intensities  of  the  shearing-stresses  on  two  planes  at  right  angles 
to  each  other  are  equal.  If,  upon  applying  this  test,  we  find 
that  the  conditions  are  not  inconsistent,  we  may  proceed  as 
follows  :  — 

Suppose  CD  (Fig.  304)  were  the  given  plane,  and  OR  the 
stress  upon  it,  and  suppose  the  position  of  the  principal  axes, 
OX  and  O  Y,  and,  indeed,  all  the  rest  of  the  figure,  were  absent, 
i.e.,  not  known.  Now,  we  can  easily  draw  the  normal  ON ' ; 
and,  if  we  could  determine  upon  it  the  point  M  such  that  OM 


CONVERSE   OF   THE  ELLIPSE   OF  STRESS.  885 

should  be  one-half  the  sum  of  the  principal  stresses,  we  should 
be  able  to  reproduce  the  whole  figure.  Hence  we  will  devote 
ourselves  to  the  determination  of  the  position  of  the  point  M. 

Let  OR  =  p  =  stress  on  plane  CD. 

Let  stress  on  the  other  given  plane  be/x. 

Let  NOR  =  0  —  obliquity  of  /. 

Let  0,  —  obliquity  of  /,. 

Then  we  have 

MR2  =  OR2  +  OM2  -  2OM.  OficosO; 

or,  if  <rx  and  <ry  denote  the  (unknown)  magnitudes  of  the  prin- 
cipal stresses, 


From  the  triangle  constructed  in  the  same  way,  with  the 
stress  on  the  other  plane,  we  should  have 


Hence,  by  subtraction, 

_i_  „  \ 

\(pcosO  — /,cos0,)  (3) 

(4) 


2  2(/COS0   —  ^COstf,) 

Having  thus  found  -        — ,  we  can  next  find,  from  either 

(i)  or  (2),  the  value  of 

Now,  therefore,  we  know  OM  and  MR,  and  hence  we  can  lay 
off  this  value  of  OM,  and  complete  the  triangle  OMR ;  then 


SS6  APPLIED  MECHANICS. 

bisect  the  angle  NMR,  and  the  line  MS  is  parallel  to  the  axis 
of  greater  principal  stress.  Hence  draw  O  Y  parallel  to  MS, 
and  OX  perpendicular  to  OY,  and  lay  off  on  OY 


=  o-y  =  OM+  MR, 
and  on  OX 

OA  —  ax  =  OM—MR, 

and  the  problem  is  solved. 

§  294.  Rankine's  Graphical  Solution.  —  The  following  is 
Rankine's  graphical  solution  of  the  preceding  problem  : 

Draw  the  straight  line  ON,  Fig.  305,  and  then  lay  off  angle 
NOR  =  0,  and    angle   NOR, 
=  0lt   also  OR=p  and   OR,    o 
=  /!.     Then  join  R  and  R,, 
and   bisect  RR,  by  a  perpen- 
dicular SM.     From  the  point 
My  where  this  meets  ON,  draw  MR  and  MR,  .     Then  will 

(i) 


.'.  (Tx  =  OM-  MR,    (3)     and  <7y  =  OM  +  MR  ;       (4) 

and  the  angles  made  by  0  Y  with  ON  and  ON, ,  respectively, 
will  be 

NMR          t  ,  o  NMR, 

9o°-a  =  — — ,        (5)     and    90    -<x,=     — — . 

A  comparison    of  this  figure  with   the  triangle   OMR  of 
§  291  will  show  that  this  is  merely  a  graphical  construction  for 


KANKINE'S  GRAPHICAL   SOLUTION. 


887 


the   analytical  solution   given  in  §  293  ;  and  the  equations  of 
that  article  can  readily  be  deduced  from  the  figure  given  above. 


SPECIAL    CASES. 

(a)  When  the  two  given  planes  are  at  right  angles  to  each 
other. — In  this  case  the  tangential  components  of  OR  and  ORl 
are  equal,  and  hence  (Fig.  306)  aR1  =  bR. 

Kence  the  figure  be- 
comes that  shown  where 
RR,  is  parallel  to  ON]  and 

if  we  let  0J  =  A,  0*  ==/,,, 

and  bR  —  aRl  —  pt ,  we  shall 
have 


<T«  —  CT, 


whence  we  readily  obtain  <rx  and  (ry,  and  then  a  and  al ,  just  as 
before. 

(b)  When  the  two  given  stresses  are  conjugate  (see  §  285). — In 
this  case  the  obliquities  of  the  two  stresses  are  the  same,  and 
the  figure  becomes  Fig.  307. 

We  then  obtain  M 


FIG. 


307- 


/i  > 
2  COS  v 


888 


APPLIED  MECHANICS. 


l--^  =  MX=  V 

2 


=  /(ILIA.) 


tan>  .  = 


(t:)  The  following  proposition,  due  to  Rankine,  will  next 
be  proved : 

The  stress  in  every  direction  being  a  thrust,  and  the  great- 
est obliquity  being  given,  it  is  required  to  find  the  ratio  of  two 
conjugate  thrusts  whose  common  obliquity  is  given. 

Let  0  denote  the  greatest  obliquity,  then  we  shall  have  (see 
last  equation  of  §292) 


Now  let  0  denote  the  common  obliquity  of  two  conjugate 
stresses  whose  intensities  are/  and/j .  Moreover,  0  <  0,  and 
we  will  consider/!  < /.  Then  from  equations  (9)  and  (10)  we 
deduce 


_ 
' 


,  cos2  0 


and  combining  this  with  (n)  we  obtain 


RANKINGS  GRAPHICAL    SOLUTION. 


//—/A2  _  cos8  6  —  cos2  0 

' 


P\  _  cos  ^  ~~  V  cos2  0  —  cos2  0 
>  ~"  cos  B  4-  l^co?"^-  cosa~0' 

wnich  is  the  ratio  shown. 

§  295.  Case  of  any  Stresses  in  Space.  —  In  the  case  of 
stress  which  is  not  all  parallel  to  one  plane,  we  should  find  that 
it  is  always  possible,  no  matter  how  complicated  the  state  of 
stress  in  a  body,  to  find  three  planes  at  right  angles  to  each 
other  on  which  the  stress  is  wholly  normal,  these  being  the 
principal  stresses  ;  and  a  number  of  propositions  follow  analo- 
gous to  those  for  stresses  all  parallel  to  one  plane.  The  discus- 
sions of  these  cases  become  very  complex,  and  will  not  be 
treated  here. 

§  296.  Some  Applications.  —  The  following  are  some  of 
the  practical  cases  which  require  the  theory  of  elasticity  for 
their  solution. 

§297.  Combined  Twisting  and  Bending.  —  This  is  the 
case  very  generally  in  shafting,  as  the  twist  is  necessary  for 
the  transmission  of  power,  and  the  bending  is  due  to  the  weight 
of  the  pulleys  and  shafting,  and  the  pull  of  the  belts,  this  being 
especially  so  when  there  are  pulleys  elsewhere  than  close  to  the 
hangers  ;  also  in  overhanging  shafts,  in  crank-shafts,  etc. 

Thus  far  we  have  no  tests  of  shafting  under  combined 
twisting  and  bending,  and  therefore  the  methods  used  for 
calculating  such  shafts  vary.  With  many  it  is  the  practice 
to  compute  their  proper  size  from  the  twisting-moment  only, 
but  to  make  up  for  the  bending  by  using  a  large  factor  of 
safety,  the  magnitude  of  this  factor  depending  upon  how  much 


890  APPLIED  MECHANICS. 


the  computer  imagines  the  shaft  will  be  weakened  by  the  par- 
ticular bending  to  which  it  is  subjected. 

With  others  it  is  customary  to  compute  the  deflections, 
under  the  greatest  belt-pulls  that  can  come  upon  it,  by  the 
principles  of  transverse  stress,  without  any  reference  to  the 
torsion,  and  to  so  determine  it  that  the  deflection  computed  in 
this  way  should  not  exceed  Y^O-  or  T61o"o  °^  tne  sPan- 

On  the  other  hand,  Unwin  and  some  others  give  the  for- 
mulae, which  will  be  developed  here  for  combined  twisting  and 
bending,  as  deduced  by  the  theory  of  elasticity.  This  formula 
has  not,  as  yet,  been  very  extensively  used ;  and  its  constants 
are  taken  from  experiments  on  tension  or  torsion  alone,  and 
not  on  a  combination  of  the  two.  It  is  to  be  hoped  that  we 
may  some  time  have  some  experiments  on  such  a  combination. 
We  will  now  proceed  to  deduce  a  formula  for  the  greatest  in- 
tensity of  the  stress  at  any  point  of  the  shaft. 

For  this  purpose 

Let  Ml  =  bending-moment  at  any  section. 

M2  =  twisting-moment  at  the  same  section. 

Il    =  moment  of  inertia  about  neutral  axis  for  bending. 

72    =  moment  of  inertia  about  axis  of  shaft. 

r     =  distance  from  axis  to  outside  fibre. 

Then,  if  we  denote  by  a-  the  greatest  intensity  of  the  stress 
due  to  bending,  and  by  T  the  greatest  intensity  of  the  stress  due 
to  twisting,  we  have, 

MS  .  M2r  ,  . 

<r  =       '  (i)  T  =  -J-.  (2) 

Jl  Jf 

For  a  circular  or  hollow  circular  shaft, 

/a=2/i; 

hence 

"-^         (3)  T  =  ^T-        (4) 


COMBINED    TWISTING  AND  BENDING.  891 

Then,  at  a  point  at  the  outside  of  the  shaft  in  the  section 
under  consideration,  we  shall  have,  — 
i°.  On  a  plane  normal  to  the  axis, 

(a)  a  normal  stress  <r, 

(b)  a  shearing-stress  T. 

2°.  On  a  plane  in  the  direction  of  the  axis, 

(a]  a  normal  stress  o. 

(b)  a  shearing-stress  r. 

We  thus  have  the  case  solved  in  Example  I.,  §  290. 
If,  therefore,  the  greatest  and  least  principal  stresses   be 
denoted  by  o-,  and  o-2  respectively,  we  shall  have 


(5) 


"  =  -2-\^+^ 

But,  if  cr  and  «2  denote  the  strains  in  the  directions  of  the 
principal  stresses,  we  have 


m  m 

Hence,  substituting  for  o-,  and  o-2  their  values,  we  have 
m  —  i 


y'  \ 
(7) 


—    I  W    4-    I    ,  -  /Qv 

—  O-    --  —  Vo-2    +    AT2.  (8) 


We  then  have,  for  the  greatest  stress  on  any  fibre,  the 
greater  of  the  two  quantities  (7)  and  (8);  and  this  should  not  at 
any  section  of  the  shaft  exceed  the  working-strength  of  the 
material  for  tension. 


892  APPLIED   MECHANICS. 

The  greater  of  the  two  is  E^  :   hence  we  should  have,  if 
f  =  greatest  stress, 

m  —  i          m  -f-  i  / 

2W  2/#  ^*' 

If,  now,  we  let  m  =  4,  as  is  commonly  done,  we  have 

O  '        8  T"  •''  \          / 

this  being  the  formula  given  by  Grashof  and  others  for  com- 
bined twisting  and  bending. 

On  the  other  hand,  Rahkine  puts  the  value  of  o-,  in  (5)  equal 
to  ft  and  hence  Rankine's  formula  is 


(T  I    / 

22 

This  might  be  derived  from  (9)  by  making  m  =  oo  instead 
of  m  =  4. 

The  formulae  developed  above  are  applicable  to  any  section. 

APPLICATION   TO   CIRCULAR   AND   HOLLOW   CIRCULAR   SHAFTS. 

Substituting  for  o-  and  r  in  (10)  the  values  from  (3)  and  (4), 
we  should  obtain 


(12) 

which  is  Grashof's  formula,  and  is  given  by  Unwin  and  others ; 
and,  substituting  in  (11)  instead,  we  should  have 

(13) 

Equation  (12)  is  equivalent  to  the  following  rule:— 


HOLLOW  CYLINDERS  SUBJECTED    TO  PRESSURE.      893 

Calculate  the  shaft  as  though  it  were  subjected  to  a  bending- 
moment 


and  equation  (13)  is  equivalent  to  the  following  rule  :  — 

Calculate  the  shaft  as  though  it  were  subjected  to  a  bending- 
moment 


ivl  I    / 

M0   = h    -V^/i 


Now,  if,  as  is  usually  the  case,  the  section  where  the  great- 
est bending-moment  acts  is  also  subjected  to  the  greatest 
twisting-moment,  it  will  only  be  necessary  to  put  for  Ml  the 
greatest  bending-moment,  and  for  M2  the  greatest  twisting- 
moment. 

§  298.  Thick  Hollow  Cylinders  subjected  to  a  Uniform 
Normal  Pressure.  —  Let  inside  radius  =  r,  outside  radius  — 
rlt  length  of  portion  under  consideration  =  unity,  intensity  of 
internal  normal  pressure  =  P,  of  external  normal  pressure 

=  P* 

i°.  Divide  the  cylinder  into  a  series  of  concentric  rings  ; 
let  radius  of  any  ring  be  p,  and  thickness  dp,  these  being  the 
dimensions  before  the  pressure  is  applied. 

Let  p  become  p  +  £,  and  dp,  dp  -f-  d£,  after  the  pressure  is 
applied. 

Then  at  any  point  of  this  ring  we  shall  have,  for  the  strain 
in  the  radial  direction, 

7'  <'> 

dp 

and,  since  the  length  of  the  ring  before  the  application  of  the 
pressure  is  2-n-p,  and  after  is  2ir(p  +  £),  hence  the  strain  in  a 
direction  at  right  angles  to  the  radius  is 

£  o 


894  APPLIED  MECHANICS. 

2°.    Impose,  now,  the   conditions  of  equilibrium   upon   the 
forces  exerted  by  the  rest  of  the  cylinder  upon  the  upper  half- 
ring.     For  this  purpose  let 
p  =  intensity  of  normal  pressure  on  inside;  i.e.,  at  dis- 

tance p  from  the  axis. 
p  -\-  dp  =  intensity  of  normal  pressure  on  outside  ;  i.e.,  at  dis- 

tance p  +  dp  from  the  axis. 
Then  we  shall  have  for  these  forces,  — 

(a)  Upward  force  due  to  internal  pressure, 


(b)  Downward  force  due  to  external  pressure, 

2(p  +  dp)(p  +  £  +  dp  +  <#). 

(c)  Upward  force  at  right  angles  to  radius  acting  at  division 
line  between  the  two  half-rings, 

2t(dp   +   <#), 

where  /  =  intensity  of  hoop-tension  per  square  unit  ;  i.e.,  of 
tension  in  a  circumferential  direction.     Then  we  have 


2(/  +  dp)  (p  +  £  +  dp  +  <#)  - 
and,  if  this  be  reduced,  and  the  terms 


,     and     2td% 

be  omitted,  all  of  which  are  very  small  compared  with  the 
remaining  ones,  we  shall  have 

dp      p  —  t 


Now,  the  two  stresses  /  and  /  are  principal  stresses,  since 


HOLLOW   CYLINDERS  SUBJECTED    TO  PRESSURE.       895 

there  are  no  shearing-stresses  on  these  planes.     Hence  we  have, 
from  equations  (i)  and  (2),  §  282, 


Now  eliminate  /  and  t  between  (3),  (4),  and  (5),  and  obtain 
a  differential  equation  between  p  and  £ 
Proceed  as  follows  :  — 
From  (4)  and  (5), 

=  ^«1/*   +  J\ 
m2  •  -  i\dp         mpj 
dp  _      Em2    /d*£       2_dA  __  1\ 
dp  ~  mz  —  i  \  dp2       mp  dp       mp2/ 


From  (4)  and  (5)  also, 

p  —  t         Em 


p  m  -f  i  \dp 

Hence,  substituting  in  (3),  and  reducing,  we  obtain 

£f  +  J  J  _  1          =o  (6) 

up          p  up        p2 


Hence,  by  integration, 


2a  being  an  arbitrary  constant,  to  be  determined  from  the  con- 
ditions of  the  problem. 


896  APPLIED  MECHANICS. 

From  (7)  we  obtain 

<#   , 

+  t==2a       or 


Hence,  integrating,  we  have 

£P  =  ap2  +  b,  (8) 

b  being  another  arbitrary  constant. 
From  (8)  we  obtain 


-,  (9) 

p 


which  gives  us,  for  the  two  strains, 


-         =   «+-,  (I!) 

Hence,  substituting  these  values  in  (4)  and  (5),  and  solving 
for/  and  t  successively,  we  obtain 

Em  Em     b  , 

p  =  -  a  ---  ,  (12) 

*        m  -  i         m  +  i  p2 

Em  Em     b  ,     x 

/=  -  a  H  --  •  --  .  (13) 

m  —  i          m  +  i  p2 

Now,  to  Determine  #  and  <^,  we  have  the  conditions,  that, 
when  p  =  r,  p  ==  P,  and  when  p  =  rlt  #  =  —Pi- 
Hence 

z>  _     Em       _      Em      b_      „  Em       _      Em      b_ 

m  —  i         m  +  i  r2'  m  —  i         m  •+•  i  r,*' 


.      a  =  ^  _  . 

£OT        ^,!  —  r2  .£«    r,s  —  ra2 

.          _  /jr.2  -  /V2         i   (P,  -  P)r'rf 


HOLLOW  CYLINDERS  SUBJECTED    TO   PRESSURE.      897 

The  greatest  value  of  /,  and  hence  the  greatest  intensity  of 
the  hoop-tension,  occurs  when  p  =.  r ;  and  hence  we  obtain 


this  value  of  /  being  negative  when  there  is  hoop-tension,  be- 
cause the  signs  were  so  chosen  as  to  make  /  positive  when 
denoting  compression.  • 

If  Pl  —  o,  i.e.,  if  there  is  no  external  pressure,  we  have 


Max/=    - 


and,  according  to  Professor  Rankine's  method,  we  should  deter- 
mine the  proper  dimensions  by  keeping  max  /  within  the  work- 
ing-strength of  the  material.  , 

On  the  other  hand,  if  we  decide  that  we  will  keep  the  value 

of  E(  -  }  within  the  working-strength,  we  shall  find  for  this,  when 
we  make  p  =  r, 

[2/X2  -  P(r*  +  r2)]  -  -P(r?  -  r2) 


o 


and,  if  m  =  4, 


When  P,  =  o, 

„ 

Max^l-  )  =  -  —  (19) 

2          * 


Practical  cases  of  thick,  hollow  cylinders  subjected  to  a  uni 
form  normal  pressure  occur  in  hydraulic  presses  and  in  ord- 
nance. 


898  APPLIED  MECHANICS. 

§  299.  Rankine's  Theory  of  Earthwork. — For  a  mass  of 
earth  bounded  above  by  a  plane  surface,  either  horizontal  or 
sloping,  his  theory  assumes  the  following  proportions  to  be 
true,  viz. : 

"  i°.  The  pressure  on  a  plane  parallel  to  the  upper  plane  sur- 
face (which  may  be  called  a  conjugate  plane)  is  vertical,  and 
proportional  to  the  depth. 

"  2°.  The  pressure  on  a  vertical  plane  is  parallel  to  the  up- 
per plane  surface,  and  conjugate  to  the  vertical  pressure. 

"  3°.  The  state  of  stress  at  a  given  depth  is  uniform." 

If  we  let  w  denote  the  weight  of  the  earth  per  unit  of 
volume,  x  the  depth  of  a  given  conjugate  plane  below  the 
surface,  0  the  inclination  of  the  conjugate  plane,  then  is  the 
intensity  of  the  vertical  pressure  on  the  conjugate  plane 

p  =  wx  cos  6.  (i) 

Moreover,  if  0  is  the  angle  of  repose  of  the  earth,  then  is 
0  the  angle  of  greatest  obliquity.  If  now  we  denote  by/  the 
pressure  against  the  vertical  plane,  then  we  have  that 


A  ^  „  cos  0  —   V  cos2  6  —  cos2  0 

f  x>  WX  COS  U — , 

cos  6  -j-  V  cos2  B  —  cos2  0 


£ 

wx  cos  B 


/-\ 


cos  6-    V  cos2  8  -  cos2  0  ' 

i.e.,  /  may  have  any  value  between  these  two. 

When  the  problem  is  to  determine  the  pressure  exerted  by 
a  mass  of  earth  against  the  vertical  face  ab  of  a  retaining-wall, 
we  determine  the  intensity  of  the  pres- 
sure p  against  the  face  (acting  along  cd) 
at  a  depth  x  below  the  upper  surface  ac 
by  means  of  equation  (2)  ;  inasmuch  as 
Rankine  claims  to  prove  by  means  of 
Moseley's  principle  of  least  resistance 
that  the  pressure  against  the  vertical  FlG-  308. 

plane  is  the  lesser  of  the  two  conjugate  pressures. 


RANKINGS    THEORY   OF  EARTHWORK.  899 


Hence,  for  the  entire  pressure  against  the  wall  we  have  a  dis- 
tributed pressure  whose  intensity  is  zero  at  a,  and  which  varies 
uniformly  as  we  go  downwards,  the  direction  of  the  pressure 
being  parallel  to  the  upper  surface  ae,  and  the  point  of  appli- 
cation of  the  resultant  being  at  a  depth  below  a  equal  to  \(ab\ 

When  the  upper  surface  ae  is  horizontal,  we  have  cos  6  =  i  • 
and  hence  (2)  becomes 

i  —  sin  0 

*-w*r+ifc0'  (4) 

and  (3)  becomes  /  =  wx  *  _  ^  ^ .  (5) 

SUPPORTING  POWER  OF  EARTH  FOUNDATION  ACCORDING  TO  RANKINE. 

Let  the  surface  of  the  ground  be  horizontal. 

Then  Rankine  says  that  the  conjugate  pressure  may  be  in- 
creased beyond  the  least  amount  by  the  application  of  some 
external  pressure,  as  the  weight  of  a  building  founded  upon 
the  earth  ;  that,  in  this  case,  the  conjugate  pressure  will  be  the 
least  which  is  consistent  with  the  vertical  pressure  due  to  the 
weight  of  the  building,  and  if  that  conjugate  pressure  does 
not  exceed  the  greatest  conjugate  pressure  consistent  with  the 
weight  of  the  earth  above  the  same  stratum  on  which  the 
building  rests,  the  mass  of  earth  will  be  stable. 

Moreover,  the  greatest  horizontal  pressure  at  the  depth •  x, 
consistent  with  stability,  is 

i  +  sin  0  , 

p  =  wx  —     —. — -  .  (6) 

i  —  sin  0 

The  greatest  vertical  pressure,  consistent  with  this  horizontal 
pressure,  is 

i  -f-  sin  0  /  i  -j-  sin  0V 

p'=p-     —r-~  =  wx(-        .     U   ;  (7) 

i  —  sin  0  \  i  —  sin  0/ 

and  this  is  the  greatest  intensity  of  the  pressure  consistent 
with  stability,  of  a  building  founded  on  a  horizontal  stratum 
of  earth  at  a  depth  x,  the  angle  of  repose  being  0. 


900 


APPLIED    MECHANICS. 


§  300.  Strength  of  Flat  Plates.  —  In  this  regard,  the  for- 
mulae that  will  be  deduced  are  those  of  Professor  Grashof,  the 
reasoning  followed  being  substantially  that  given  by  him  in  his 
"  Festigkeitslehre." 


ROUND    PLATES. 

Let  the  curved  line  CA  be  a  meridian  curve  of  the  middle 
layer  of  the  plate  after  it  is 
bent.  Take  the  origin  at  O  ; 
let  axis  OZ  be  vertical,  and  axis 
OX  horizontal,  and  let  the  axis 
at  right  angles  to  ZOX  be  <9<£, 
so  that  zt  x,  and  <j>  are  the  co- 
ordinates of  any  point  in  the 
middle  layer  of  the  plate. 

Let  y  denote  the  (vertical) 
distance  of  any  horizontal  layer 
from  the  middle  layer  of  the 
plate. 

Let  R  =  radius  of  curvature 
of  meridian  line  at  any  point 

OF,  0).  FIG.  300. 

Let  Rl  =  radius  of  curvature  of  section  of   middle  layer 
normal  to  meridian  line. 

Then  we  should  have,  from  the  differential  calculus, 


I 
R 


dx* 


('  + 

\ 


\dx 


d*z 

-  ^  nearly> 


i  i  dz 

R^  x  dx' 

Hence,  reasoning  in  the  same  way  as  in  the  common  theory  of 
beams,  we  should  have,  for  the  strains  of  the  layer  whose  dis- 


STRENGIft   OF  FLA  T  PLA  TES.  QOI 

tance  from  neutral  layer  is  y  at  point  (x,  z],  provided  there  is 
no  stress  in  the  plane  of  the  neutral  layer, 

y  y 

<-  =  ±  y  V  =  ±  £• 

When   there  is   such  a  stress,  let  the  strains  due  to  that 
stress  be  e*0  and  e$0. 
Then  we  shall  have 


dz 


Hence,  substituting  in  (i)  and  (2)  of  §  283,  we  have 


d2z    .    m  dz 


Now  let  us  suppose  the  plate  to  be  subjected,  before  load- 
ing, to  a  uniform  pull  in  its  own  plane,  and  normal  to  its  cir- 
cumference ;  and  let  the  intensity  of  this  pull  be/x.  Then 


and  hence,  we  have, 


Therefore,  substituting  in  (3)  and  (4),  and  reducing, 

mEy   I    d2z        i  dz\ 

<rx  =  A (  m- h  -  -r  )'  (6) 

m2  —  i\    ^c2       ^  d$l 

z        m  dz' 


902  APPLIED   MECHANICS. 

These  equations  express  the  stresses  in  terms  of  the  co- 
ordinates of  the  points. 

Now  impose  the  conditions  of  equilibrium  upon  the  forces 
acting  on  any  half-ring  of  thickness  dx  =  d$. 
These  forces  are  — 

i°.  Force  exerted  upon  it  by  the  outer  part 
of  the  plate, 

\2X<TX  -f    2d(X(T x)  I dz. 

2°.  Force  exerted  by  the  inner  part  of  the  plate, 

—  2XVxdz. 

3°.  Force  exerted  upon  it  by  the  other  half-ring, 


4°..  Force  exerted  by  resistance  to  shear  on  top  and  bottom, 

\  (T  +  dr)  —  T  1 2xdx. 

Hence,  equating  to  zero  the  algebraic  sum  of   these,  and 
reducing,  we  obtain 

dr        dr        <TA>        i  d(x<Tr\ 


dz       dy        x        x      dx 


(8) 


Now  substitute  for  <rx  and  0-4,  their  values,  and  reduce,  and 
we  have 

dr         m2Ey  fd^z        i  d2z        i   dz  \  ,  . 

-  —  =  -        — (  —  -\ i.  ^9 ) 

dy       m2  —  i  \dx3       x  dx2       x2  dx/ 

Integrate  with  regard  to  j,  and  we  have,  since  the  quantity 
in  brackets  is  not  a  function  of  y, 

fd*z       i  d2z       i  dz 
\dx*       x  dx2       x2  dx 


STRENGTH  OF  FLAT  PLATES.  903 

But,  when^/  =  -  (k  being  the  thickness  of  the  plate),  T  =  o, 
since  there  is  no  shearing-force  at  top  or  bottom  ; 

m2Eh2     /d*z        i  d*z        i  dz 
~  S(m2  -  i)V&3       xdx*       x* 


m*E(h*  -  *f)(d*z    .\d-z_   JE_  dz\ 
S(m2  -  i)     W        *  afc*       *2  <&/' 


This  gives  us  the  intensity  of  the  shearing-stress  at  any 
point  (;r,  z}  at  distance  7  from  middle  layer  ;  and  this  is  the 
intensity  of  the  shear  at  that  point  between  two  horizontal 
layers,  and  hence  also  along  a  vertical  plane  through  the  point 

fo  *)• 

Now  let  us  take  the  case  of  a  centre  load  P  combined  with 
a  distributed  load  p  per  unit  of  area.  Then  shearing-force  at 
distance  x  from  centre  = 

TTX2P    +    P, 

this  tending  to  shear  out  a  circular  piece  of  radius  x.  Hence 
we  must  have  this  balanced  by  the  whole  shearing  resistance 
on  the  surface  subjected  to  shear; 


Now  substitute  the  value  o£  T  from  equation  (TO),  integrate, 
and  reduce,  and  we  obtain 

d*z        i  d2z        i   dz  6(m2  — 

dx*       x  dx*       x2  dx 


904  APPLIED   MECHANICS. 

Hence,  for  the  intensity  of  the  shearing-force,  we  have 


This  gives  the  intensity  of  the  shearing-force  at  any  point  of 
the  plate. 

Next,  to  find  its  deflection,  or  the  equation  of  the  meridian 
line,  we  have,  from  (12), 


,    £^\  6(^2  -  i)/  P 

2       x  dx)  m2Efc     V          TTX 


dxdx2 


x  dx  m2Eh*        2 

d2z    ,    dz  6(m2  -  i)    x*        6(m2  -  i)  P 


__  6(^2  -  i)  P 

°^** 


But 


hence,  integrating,  we  have 
dz  _        6(m2  -  i)   *4 


6(m2  -  i)  T'^2  6(m2  -  i) 

+        ^3         ^ 


Hence,  dividing  through  by  x,  and  integrating, 

6(m2  —  i)   x4 

2  = i p— 

32 

-  i)  Px2,,  ,   ex* 


TT  4  4 


and  this  is  the  meridian  line  of  the  surface,  the  constants  c.  a* 
and  e  being  as  yet  undetermined. 


STRENGTH  OF  FLAT  PLATES.  905 

This  is  as  far  as  we  can  proceed  before  taking  up  special 
cases. 

(a)  Full  Plate.  —  When  the  plate  is  full,  the  slope  becomes 
zero,  for  x  =  o ;  therefore  (14)  gives  us 

d  =  o, 
and  in  this  case  (15)  becomes 

6(m*  -  i)   ** 

-p— 

32  • 


(a)   Unifdrmly  Loaded,  no  Centre  Load.  —  P  =  O 
6(m2  —  i)    x*       ex2 


But  when  .*•  =  ^,  #  =  o  ; 


And,  substituting  for  #,  —  and  —  —  ,  their  values  in  (i)  and  (2), 

£ZJv  ^Z^v 

we  obtain 

—  i  ^/3  6(w2  —  i)  A 

•  *»  -      *    <'9> 


Et    =  m  ~  V     +    "(*  6(m*  ' "  J) 
and  (13)  gives 


906  APPLIED    MECHANICS, 


(ft)   Supported  all  around.  —  When  x  —  ry  vx  =  O-XQ  =  ^>,  for 
all  values  of  y:  therefore,  from  (6), 

\ldz 
r\dy. 

yVpp*  /7^^^ 

and,  substituting  the  values  of  -  -  and  —  as  determined   by 

dx          dx* 

differentiating  (18),  we  have,  after  reducing, 

4-  i)  pr* 


c  — 

"2  m2  Eh* 


Hence  equation  of  meridian  line  is 

r2_ 


m  4-  i 

Hence  we  have  maximum  deflection  by  making  x  =  o  ; 

3    (m  -  i)(5»g  +  i)  pr* 

~  -'  (23) 


And,  substituting  in  (19)  and  (20),  we  obtain,  after  reduction, 
£,          m  ~  i  3  w2  —  i  /  (3^  +  i  ) 

^  =  ~^~A  +  4  ^^  ^l^T7r    -  3*  p  (24)  . 


_  ^          .  (25) 


m  4       w2      /^3(  w  -f  i 

But,  in  a  plate  supported  all  around,  /,  =.  o  ;  and  then  the 
maximum  value  of  either  one  occurs  when  y  =  -,  and  hence 


. 


STRENGlIf  O*  SLAT  PLATES.  907 

On   the   other  hand,  T  becomes  greatest  when  x  =  r  and 
y  =  o.     Hence 

3  r 
Maxr  =  -  -,p; 

4  /f 

and,  if  eT  represent  the  maximum  strain  due  to  this  shearing- 
force,  we  have 


RESULTING  FORMULA  FOR  PLATE  SUPPORTED  ALL  ROUND. 

Max  EzQ  =  —  p          or p 

8  tn2  h2  ^       m      h 

whichever  is  greatest. 

0o  =  —  —  — . 

10  tn2  £,n* 

PLATE   FIXED   AT  ENDS. 


Equation  (17)  applies  to  this  case  also. 

Now,  when  x  =.  r,  —  =  o ; 
dx 


=  A  m*  ~  l  ^(r*  -  ««)•.  (28) 

1  6       w2  v 


Hence  greatest  deflection  is 


=  __ 
1  6 


APPLIED   MECHANICS. 


and 


When  /,  is  positive  or  zero,  then  E*x  is  maximum  for  x  =  o 
y  =  -,  and   for  x  =  r,  j  =   --  ;   and  E^  is    maximum  for 

x  =  o,  _y  =  -  :  and  the  maximum  value  of  .ZTe$  is  equal  to  first 
maximum  of  Etx.     We  have 


,  „  /«  —  i  T.  m2  —  \  r2 

Second  max^c^  =  -      —  /,  -f  -      —  -  --  -/.  (32) 

Hence  the  second  is  the  real  maximum. 


RESULTING  FORMULAE   FOR   PLATES   FIXED   AT   THE  ENDS. 
m  —  i  *  m2  —  \  r2  L 


1  6       w2 
For  pl  =  o, 

TiT       r^  <?  w2  —  i  r2 

Max^€o  =  ^-  -  -/. 

4       m2      h2 

§  301.  Thickness  of  Plates.  —  Grashof  advises  the  use  of 
3  as  value  of  m.  If  this  be  adopted,  we  should  have,  for  the 
proper  thickness  of  round  plates, 

Supported.  Fixed. 


*  =  rV£?,  h  =  rV- ; 


RECTANGULAR   PLA  TES. 


909 


where  h  =  thickness,  r  =  radius,  /  =  pressure  per  square  inch, 
and  /  =  working-strength  per  square  inch.  If,  now,  we  use  a 
factor  of  safety  8,  and  use  as  tensile  strength  of  cast-iron  20000, 
of  wrought-iron,  48000,  and  of  steel  80000,  we  should  have :  — 


Supported. 

Fixed. 

Cast-iron  .     .     . 
Wrought-iron 
Steel     .... 

h  =  0.0182570;-^ 
h  =  0.01178507-^ 
h  =  0.0091287^^ 

h  =  o.oi633OO?V/ 
h  —  o.oio54iory/ 
h  =  0.008  1  649  r  v5> 

§  302.  Rectangular  Plates.  —  Refer  the  plate  to  rectangu- 
lar axes,  as  before,  OZ,  OX,  O& ;  the  origin  being  at  the  middle 
of  its  middle  layer. 

Let  y  —  distance  of  any  point  in  the  plate  from  the  middle 
layer. 

Let  px  be  the  radius  of  curvature  of  a  normal  section  par- 
allel to  OX  at  the  point  (x,  z,  <f>). 

Let  p<f>  be  the  radius  of  curvature  of  a  normal  section  par- 
allel to  O$  at  the  point  (x,  s,  </>). 

Then  we  shall  have,  by  the  principles  of  the  common  theory 
of  beams, 

=  c    ±  L 

0     P* 


<x  =  e,o  ±        , 

P* 


where  *x    and  ^  are  the  strains  of   the  middle  layer  in   the 
directions  OX  and  O®  respectively. 

Moreover,  from  the  Differential  Calculus,  we  have 


—  COS2  A.   -f    2 —  COS  A  COS  fJi    H-   — -COSa/X 

dx* 


910  APPLIED   MECHANICS. 

where  A  =  angle  between  normal  and  z  axis,  and  ^  =  angle 

between  normal  and  x  axis.     But  --  and  —  being  the  slopes, 

dx         d<$> 

and  hence  small,  we  shall  have  nearly 

cos  A  =  i,        cos /A  =  o, 
j[_  d2z  i  d2z 

d2z 


(2) 


Hence  (i)  and  (2)  of  §  283  give  us 
mE  mE 


m2  - 


And,  if  cr^,  a-^  ,  denote  the  stresses  in  the  middle  layer,  we 
shall  have,  since 

.  mE  , 


Now,  if  ^  and  rj  denote  the  increments  in  x  and  <£  respec- 
tively due  to  the  load,  we  shall  have 

dz  d£  d*z 


RECTANGULAR  PLATES.  9!  I 


But 


hence 


Equations  (3),  (4),  and  (5)  are  the  expressions  giving  the 
stresses  on  two  planes  at  right  angles  to  each  other,  parallel  to 
OX  and  ^respectively.  Hence  we  have  a  case  of  stress  on 
two  planes  at  right  angles  to  each  other,  and  we  are  to  find  the 
principal  stresses  :  we  thus  have  — 

i°.  Normal  stress  on  x  plane,  vx. 

2°.  Shearing-stress  on  x  plane,  -r^. 

3°.   Normal  stress  on  <£  plane,  <r^. 

4°.  Shearing-stress  on  <£  plane,  T^ 

Hence,  if  we  denote  by  o-,  and  o-2  the  maximum  and  mini- 
mum  principal  stress,  we  have  (§  290) 


(6) 


(7) 

and  hence,  if  e,  and  €2  denote  the  strains  in  the  directions  of 
the  principal  stresses, 


o-2        m  —  i 

-  —  (<*x  T  °"<f>) 

m  2m 

m 


,      (8) 


m  —  i 


(9) 


912  APPLIED   MECHANICS. 

and  for  the  strain  e3,  parallel  to  OZ,  we  have 

<TX    -f    OVfr 

E^  =  --  ST--  <'°> 

In  order  to  use  (8),  (9),  and  (10),  however,  we  must  know 
°".r,  °>>  and  TJ^  ;  and  for  this  purpose  we  must  know  the  equa- 
tion of  the  middle  layer  after  bending.  For  this  purpose,  apply 
the  equations  (i),  (2),  (3),  of  §  281  to  any  particle  dxd$dz  in 
the  interior  of  the  body.  We  have  then,  X  =  Y  =  Z  =  o. 
Therefore 

dvx         drxz         drx$  _  ^       drxz  _     _/^>    , 

dx"    dy      "   d$''  ~dy   ''        \dx  "   d 


drx$         dr^z  __  .       </r^g  _         f^±    ,    ^r.rA 

dx      '   dy    '  dy    ''         \d$  "    dx  f 


x 

dx 


Therefore,  making  use  of  (3),  (4),  and  (5)  with  the  above 
conditions,  we  deduce 


mEy 


;;/ 


,      -       ,    .   > 

i  ax2a<l> 

3  xx 

\ 

/' 


=.    n?E^I<&_  _      d*z  \ 
w2  -  i  \</<£3       do?d<$>) 


RECTANGULAR  PLATES.  913 

Hence,  by  integrating  (15)  and  (16),  we  have 
TXZ  =  - — ( 1 -)  4-  clt 

m2Ev2     ld*z          d*z  \ 
2(m2  —  i)v/<£3       dx2d^>/ 

But  when  v  =  -,  r^z  —  rxz  =  o  • 

2 

m2Eh2     /d*z         d*z   \ 

-••    '«--i(2rz-'-=[  +  - 

and 


m2E 


and 

*  d^z          d^z   \f  v2 

TXZ  =  ~     (  -r-  +  -T-TT-  )( 


8 

Hence 


_  _ 
~ 


m 


Now  we  have  <rz  =  /,  where  /  is  the  intensity  of  the  load ; 
therefore  the  third  equation  gives  us,  on  integrating  between 

the  limits  -  and  —  -, 


h  k 

C-dr^         p 
J_^  ^>  J_h_ 


dx 


914  APPLIED  MECHANICS. 


m*E    \  Iv*  _  h2v\/d*z 
2  -  i(\6  ~    ~S~)\^ 


and  this  is  the  differential  equation  of  the  surface,  and  should 
be  integrated  in  each  special  case. 


INDEFINITE  PLATES  WHICH   ARE    FIRMLY    HELD    AT    A    SYSTEM    OF 
POINTS   DIVIDING   THEM   INTO   RECTANGULAR   PANELS. 

Let  the  sides  of  the  panels  be  2a  and  2b.  Assume  the 
origin  at  the  middle  of  the  panel,  the  axis  of  x  being  parallel 
to  2a,  and  the  axis  of  y  parallel  to  2b.  We  shall  in  this  case 
have  the  following  conditions  ;  viz.,  — 

(a)  —  =  o  for  x  —  ±a  and  all  values  of  <f>. 
dx 

(b)  —  =  o  for  (f>  =  ±b  and  all  values  of  x. 


(c)  z    =  o  when  x  = 

(d)  If  we  develop  the  value  of  z  in  powers  of  x  and  <£,  there 
must  enter  only  even  powers  of  x  and  <£,  since  the  value  of  z 
remains  the  same  when  we  put  —  x  for  xy  or  —  <j>  for  <j>. 

Now,  if  we  write 


=  A  +  fix2  -f  C<p  +  Dx*<p  +  Ex*  -h  F<p 

f  Lfi  -I-  Mot,  etc., 


the  above  conditions  will  be  fulfilled:  — 


RECTANGULAR   PLATES.  915 

i°.  By  making  all  the  co-efficients  after  the  fourth,  each  zero. 
2°.  By  making  D  =  o,  therefore  writing 

z  =  A  +  Bx2  4-  C<f>2  4-  Ex*  +  7^)4. 
Now 

—  =  2Bx  4-  4-Z£#3,  —  =  2  C(f>  - 


.-.      2Ba  +  4^3  =  o,  2  Cb 

and 

o  =  ^  4-  Ba?  +  C/^2 


.'.        A    = 

Hence  the  equation  becomes 

Ex* 


—  - 

&3C 


also 


=  o,  =  o,  =  o, 

*  ^^V^La  '  J+S2JA.2 


Hence  equation  of  the  middle  layer  is 

£(«•  -  ^)2  +  F(b*  -  ^>2)2,  where  E  +F=  ^  ~^P.     (18) 


APPLIED   MECHANICS. 


Now,  in  the  case  of  an  ordinary  beam  fixed  at  both  ends, 
and  loaded  uniformly  with  /  Ibs.  per  unit  of  area,  if  b  is  the 
breadth,  we  have  :  — 

i°.  The  points  of  inflection  are  at  a  distance  from  the  middle 

equal  to  —=,  where  a  is  the  half-span  ;  and 

V3 
2°.  The  bending-moment  at  a  section  at  a  distance  x  from 

the  middle  is  — f  —  —  x2 \  when  x  <  -=,  and **—  [x2  —  —  )  when 
2\3  /  V/3  2\  3/ 

x  >  —  ;  therefore  the  value  of  z  is  found  from  the  formula 
V3 

6^    (*a   fa(  „       a*\  .  a 

X  >— -=.\ 


r3 
or 


when  x  < 


Either  one,  when  integrated,  gives  for  <s-  the  value 


2  = 


Hence  in  the  flat  plate,  if  b  =  o,  the  values  of  E  and  F  must 
be  such  that  the  formula  shall  reduce  to  z  =  (a2  —  x2)2 

z 


when  b  =  o.     Now,  it  does  reduce  to  z  —  E(a2  —x2)2.     There- 
fore E  must  be  such  a  function  of  a  and  b,  that,  when  b  —  o,  it 

shall  reduce  to  -±—       So  likewise  F  must  be  such  a  function 


of  a  and  £,  that,  when  a  =  o,  it  shall  reduce  to  -—=-.     Suppose, 

2  En* 

then,  we  put 

E  =  _--  +  ^          and         ^ 


since  these  functions  fulfil  the  above  conditions. 


RECTANGULAR  PLATES.  917 


Now  we  have 


<^  ^5" 

Hence,  substituting  for  -—  and  -7-  their  values,  and  observ* 

dx2          u<l>2 

ing  that 


is  greatest  for  x  =  ±a,  y  =  ±-, 


is  greatest  for  $  =  ±£,  JF  =  ±-, 


tve  obtain 

±  2 


Ql8  APPLIED  MECHANICS. 


These  may  be  written  as  follows : 


max.  £(4.  =  <rfc  -  io>o  ±  2 


We  have  also,  by  substituting  for  E  and  F  their  values  in 
equation  (18), 

{a* ^—£n                        bn —a*  1 
^r—(a2  —  x2)2  H ^-r-(fr  —  <£2)2  I  • 
an  -j~  bn                               an  -\~  on                      ) 

In  these  results  the  exponent  n  is  undetermined,  and  we 
have  no  means  of  determining  it  in  the  general  case.  We  only 
know,  that,  since  the  deflection  must  increase  for  a  decrease  in 
x  and  <£,  therefore  we  must  have,  whenever  a  >  b, 

c«*  .-.   «<2-}o%m 


This  leaves  the  general  case  indeterminate  ;  but  a  common 
practical  case  is  not  subject  to  this  indetermination,  i.e.,  the 
case  when  a  =  bt  for  then 


er-er- 


whatever  the  value  of  n;  and  hence  equations  (21),  (22),  and 
(23)  give 

r-    \  i          ,    m2  —  i  a2  ^  ,     ^ 

max  v/V,)  =  ^  -  -<^o  ±  -^^-  ~f,  (24) 


max  Ei*  =  o>    -  l<r^o  ±  OT    ~  '  ^ 


RECTANGULAR  PLATES.  919 

Z  =  m*  ~2    I  -J^K«2  -  X2)2  +  (0*  -  <#>2)2^  (26) 

and 

m2  —  i     a* 


FORMULA  FOR  THE  SHEETS   OF  A  LOCOMOTIVE  FIRE-BOX. 

In  this    case  we   have  a  =  b  ;    hence  (24),   (25),  and  (27) 
apply  :  and  if  we  write,  with  Grashof,  m  =  3,  they  become 

max  (.EVr)  =  tr^  -  I  o*o  +  -  ^-/,  (28) 

max  (^)  =  o*o  -  i  0-^  +  -  ^/,  (39) 

(30) 


Now,  in  the  case  of  the  horizontal  sheets,  O-XQ  =  o-^  =  o, 
and  we  have 

max  (E*x)  =  -  j-j,  (31) 


In  the  case  of  the  vertical  walls,  inasmuch  as  these  have  to 
resist  the  steam-pressure  in  a  vertical  direction,  the  inner  one 
is  called  upon  to  bear  compression,  and  the  outer  tension,  in  a 
vertical  direction.  If  /  is  the  length  of  the  outside  of  the  fire- 
box, and  /,  its  breadth,  we  shall  have  for  the  outer  plate,  taking 
axis  of  x  vertical, 


92O  APPLIED   MECHANICS. 

and  for  the  inner  plate,  if  /  and  //  are  corresponding  dimen- 
sions of  inside  of  fire-box, 


And,  by  making  these  substitutions  in  (28),  (29),  and  (30),  we 
obtain  our  formulae. 


RECTANGULAR  PLATE   FIXED   AT   THE  EDGES. 

For  this  case  Grashof  deduces  the  equation  of  the  middle 
layer  as  follows  : 

i°.  This  equation  must  be  a  function  of  x  and  <f>. 

2°.  If  2a  and  2b  are  the  sides  of  the  plate,  this  function 
must  become 

(a)  When  £  =  oo  for  all  values  of  <£, 

<«*-"•>•• 


(/?)  When  a  =  oo  for  all  values  of  x, 


2  = 


because  the  plate  then  becomes  a  beam  fixed  at  the  ends. 
The  function  that  will  satisfy  these  two  conditions  is 

p      (g-  -  aa)a(j.  _  py 
a*  +  & 


From  this  he  deduces  for  max  zy  when  x  =  $  =  o, 


max  2  = 

2 


RECTANGULAR   PLATES.  Q2I 


From  (i)  he  deduces 

(3) 
(4) 
(5) 
(6) 

(7) 
(8) 

do?            Efc              a*  +  b*            ' 
d2z  _          2p    (a2  —  x2)2(b2  —  3<£2) 

d$2            Efc              a*  +  b* 
d2z          &p    (a2  —  x2)*^2  —  <f>2)<f> 

dxd<$>       Efo               a*  +  b*              ' 
d2z         Ap       a2b*             c 

tYITY              —                                                    TOT         T   -~     T~  fL        rn   —   O 

dx2        Efa  «4  -|-  b* 

d<^2       Eh2  a4  -{-  b* 

d2z          12     i)       (fifo                           i                    i  , 
max             —  °       f-     .     for    x2  —     a2.     d>a  —     0*. 

27  ^/^3  ^4  _|_  &  3  3 

these  corresponding  to  the   points   of   inflection   of  a  loaded 
beam  fixed  at  the  ends. 

Hence  (i),  (2),  and  (5)  of  §  300  give 


max 


max 


max  (r2)  = 

27 


I 

"  m*' 
m 

2a*      b2 

\yj 

(10) 
Siit 

'o    ±   ^4   +    ^4  frP> 

2a2b2     ab  ^ 

At  the  places  where  c^  and  ^  are  greatest, 


O. 


At  the  place  where  rz  is  greatest, 

O"*    =    <*x^  °">?    =    O"T)^« 


Hence  it  is  either  (9)  or  (10)  that  gives  us  the  suitable 
formula  to  use  in  any  special  case. 


Q22  APPLIED  MECHANICS. 


EXAMPLES  OF    THEORY  OP  ELASTICITY. 

1.  It  has  been  sometimes  proposed  to  use  oblique  seams  in  a  boiler- 
shell.     Assume  the  seams  at  an  angle  of  45°  with  the  axis  of  the  boiler, 
a  pressure  of  100  Ibs.  per  square  inch  of  the  steam,  and  a  diameter  of 
4  feet.     Find  the  tension  per  inch  of  length  of  seam,  and  its  direction. 

2.  Given  a  shaft  carrying  80  HP,  and  running  at  250  revolutions 
per  minute.     Suppose  the  driving-pulley  to  be  at  the   middle   of  the 
length,  this  being  6  feet,  and  given  that  the  ratio  of  the  tension  on  the 
tight  side  of  the  belt  to  that  on  the  loose  side  is  3.75.     Find  the  proper 
size  of  shaft,  assuming  10000  Ibs.  per  square  inch  as  the  working-strength 
of  the  iron. 

3.  What  should  be  the  thickness  of  a  flat  plate  to  bear  150  Ibs. 
pressure  per  square  inch,  and  stayed  at  points  forming  squares  8  inches 
on  a  side,  the  plate  being  of  wrought-iron,  working-strength  10000  Ibs. 
per  square  inch. 

4.  Find   inner  radius   of  .a  hydraulic  press  to  bear  1500  Ibs.  per 
square  inch,  given  outer  radius  =  18  inches;  material,  cast-iron;  ten- 
sile strength  20000  Ibs.  per  square  inch. 


INDEX. 


PAGE 

ACCELERATION 75 

Angular  momentum 106 

Arches 779 

conditions  of  stability 800 

correcting  joints 81 1 

criterion  of  safety 80 1 

criterion  of  stability 818 

elastic 827 

general  remarks 825 

linear 789 

line  of  resistance 79"> 

line  of  resistance  determined  by  two 

points 804 


modes  of  giving  way 

Schemer's  method 

true  line  of  resistance 

unsvmmetrical  arrangement 

At  wood's  machine 


790 
805 
80.5 
819 
79 
Axes  of  symmetry  of  plane  figures 115 

BAR-IRON,  tests  by  Kirkaldy 394 

Bauschinger,  building-stones 717 

cast-iron  columns 372 

cement 733 

repeated  stresses 531 

timber 707 

Beams   assumptions  of  common  theory.  268 

cross-section  of  equal  strength 298 

deflection  of 299 

deflection  with  uniform  bending  mo- 
ment   306 

fixed  at  the  ends 312 

load  not  at  middle 307 

longitudinal  shearing 319 

mode  of  ascertaining  dimensions.  ...  292 

mode  of  ascertaining  stresses 291 

modulus  of  rupture 293 

moments  of  inertia  of  sections 275 


PAGE 

Beams,  oak 684 

position  of  neutral  axis 269 

principles  of  common  theory 267 

rectangular,  slope  and  deflection.  ...  311 

resilience  of 306 

shearing-force  and  bending-moment .  272 

slope  and  deflection 301 

slope  and  deflection,  special  cases..  302 
slope  and  deflection  under  working- 
load 310 

spruce 675 

table  of  deflections  and  slopes 305 

table  of  shearing-forces  and  bending- 

moments 274 

timber,  strength  and  deflection 673 

timber   time  tests 688 

uniform  strength 296 

variation    of    bending-moment    with 

shearing-force 317 

white-pine 686 

working-strength 293 

wrought-iron,   strength    and    deflec- 
tion   440 

yellow-pine,  strength    and    elasticity  68 1 

Beardslee,  effect  of  rest 402 

reduction  in  rolls,  wrought-iron 401 

shape  of  specimen 400 

tensile  limit 400 

tests  of  wrought-iron 399 

Bending  and  torsion  combined 889 

and  twisting 338 

Bending-moment 185 

in  beams 272 

Bending-moments,  graphical  representa- 
tion   287 

Bollman's  truss 219 

Bow's  notation 143 

Breaking-strength 245 

923 


924 


INDEX. 


PAGE 

Bridge  columns,  Clark,  Reeves  &  Co.  ...  415 

Watertown -arsenal  tests I  421 

Sondericker's  formulae 419 

Bridge-trusses 184 

actual  shearing-force 203 

compound 208 

concentration  of  loads  at  joints 203 

counterbraces 200 

diagonals 200 

examples 186 

general  formulae 209 

general  remarks 219 

method  of  sections 180 

shearing-force   and   bending-moment  185 
steps  in  determining  stresses  under 

fixed  load 186 

vertical  posts 202 

with  vertical  and  diagonal  bracing. .  .  198 

BuiJding-stones 714 

Buttress,  stability 703 

CAST-IRON 356 

columns..  . 364 

composition  and  characteristics 356 

list  of  experimenters 358 

tension 359 

transverse  strength 375 

Catenary 784 

transformed 7*7 

Cement  mortar 720 

Centre  of  gravity 221 

Centre  of  gravity,  examples 286 

f  a  line 225 

of  a  slender  line 224 

of  flat  plate 223 

of  homogeneous  bodies 223 

of  plane  areas 224 

of  solid  bodies 226 

of  symmetrical  bodies 237 

of  system 221 

Pappus's  theorems 234 

Centre  of  percussion 1 20 

Centre  of  stress 263 

Centres  of  percussion  and  of  oscillation, 

interchangeability 123 

Chain  cable 403 

Chain  or  cord,  loaded 779 

Coefficients  of  expansion 507 

Cold  rolling. 495 

Collision 103 


PAGE 

Columns,  cast-iron 364 

Euler's  rules 326 

Gordon's  formulae 324 

Hodgkinson's  rxiles 328 

oak • 653 

strength  of 323 

timber 652 

white-pine 659 

wrought-iron 414 

yellow-pine 653,  664 

spruce 670 

timber  with  bolster 670 

Components  of  velocities  of,  and  forces 

acting  on,  a  body 82 

Compound  bridge-trusses 208 

Compression,  direct 253 

of  timber 652 

of  wrought-iron 413 

Wohler's  experiments 254 

Continuous  girders 743 

distributed  and  concentrated  loads.  .    770 

examples  for  practice 778 

loads  concentrated 763 

loads  distributed 744 

Contraction  of  area,  Kirkaldy 394 

Cord  or  chain,  loaded 779 

Cord    with    load    uniformly    distributed 

horizontally 782 

Counterbraces 200 

Couple,     composition     of,     in     inclined 

planes 59 

Couples,  composition  of 57 

effect  on  rigid  body 55 

effect    when    forces    are    inclined    to 

rod 54 

measure  of  rotary  eftect 53 

moment  of 53 

effect  on  rigid  rod 51 

representation  by  a  line 59 

resultant  with  single  force 61 

Crystalline  fracture,  Kirkaldy 498 

Crystallization  of  iron  and  steel 498 

Cylinders,  thick  hollow,  strength  of...    893 
thin  hollow 251 


DEFLECTION  of  beams. 

Domes 

Dynamics 


•    299, 301 

843 

75 


ECCENTRIC  load  on  columns.  .  .    331,  366,  420 


INDEX. 


925 


PAGE 

Elasticity,  modulus  for  timber  beams.  .  696 

modulus,  cast-iron 360,  363 

modulus  for  use  with  timber  beams.  .  695 

modulus  of 240 

modulus,  Rosset 363 

modulus,  wrought-iron 405,  407,  409 

theory  of 852 

Ely,  rules  for  strength  of  timber  posts.  .  671 

Energy 78 

Equilibrium  curves 779 

Euler's  rules  for  columns 326 

Expansion,  coefficients 507 

Ewing  cast-iron  columns 374 

Eye-bars,  steel,  Watertown  Arsenal 487 

wrought-iron 412 

FACTOR  of  safety  for  iron  and  steel 519 

timber 672 

Pink's  truss 217 

Floors,  timber 699 

Force 3 

and  momentum,  relation  between ....  1 1 
applied  at  centre  of  gravity  of  rigid 

rod 51 

applied  to  rigid  rod,  not  at  centre.  .  .  46 

centrifugal 81 

centrifugal,  of  solid  body 85 

characteristics  of 1 6 

criticism  of  definition 6 

definition  of 8 

deviating 82 

external '• 9 

intensity  of,  distributed 40 

measure  of 5,9,76 

moment  of 30 

relativity  of 9 

resultant  of,  distributed 40 

single,  at  centre  of  rigid  rod 43 

Forces,  centre  of  system  of  parallel 38 

composition  of ' 2 1,  23 

composition  of  parallel 37,  62 

composition  of  parallel,  in  a  plane.  .  .  36 

composition  of  two 30 

co-ordinates  of  centre  of  parallel.  ...  39 

decomposition  of 19 

distributed 39 

effect  of  pair  on  rigid  rod 50 

equilibrium    of 28 

equilibrium  of,  in  a  plane 69 

equilibrium  of,  in  space 73 


PAGE 

Forces,  equilibrium  of  parallel 37 

equilibrium  of  three  parallel 31 

moment,  causing  rotation 49 

normal  and  tangential  components .  .  80 

parallelogram  of 16 

polygon  of 23 

resultant  of  any  number  of  parallel.  .  35 

resultant  of,  in  a  plane. 66 

resultant  of,  in  space 70 

equilibrium  of,  in  space 73 

resultant  of  two  parallel 33 

equilibrium  of  three  parallel 31 

statical  measure  of 12 

Frames,  stability 140 

Frame,  triangular 141 

isosceles  triangular 143 

polygonal 145 

Frames  of  two  bars 138 

Frames,  stability 140 

Framing-ioints 698 

Friction  of  blocks 791 

Funicular  polygon 147 

G,  relation  to  £ 869 

G,  value  of 870 

Girder,  greatest  stresses 193 

Girders,  continuous 743 

distributed  and  concentrated  loads.  .  770 

examples  for  practice 778 

loads  concentrated. 763 

Gordon's  formulae ; 324 

Giade,  effect  on  tractive  force 100 

Gravity,  centre  of 42,  221 

HALF-LATTICE       girder:    travelling-load, 

greatest 192 

Hammer-beam  truss 176 

wind  pressure 179 

Harmonic  motion 102 

Headers  of  timber 698 

Hodgkinson's  rules  for  columns 328 

tests  of  cast-iron  columns 364 

Hooks,  strength  of 322 

IMPACT 123 

central r  24 

coefficient  of  restitution 125 

coefficient   of   restitution   by  experi- 
ment   132 

elastic 127 


926 


INDEX. 


I'AGF 

Impact ,  imperfectly  elastic 1 29 

inelastic 126 

oblique 134 

of  revolving  bodies 136 

special  cases  of  elastic 128 

Impact,  special  cases  of  imperfect  elas- 
ticity    132 

special  cases  of  inelastic 128 

velocity  at  greatest  compression.    .  .  . 

KlRKALDY. 

tests  of  bar  iron , 


LAUNHARDT'S  formula 248,  524 

Load,  sudden  application  of 246 

Longitudinal  shearing  of  beams 319 

MASS,  measiire  of 10 

unit  of .  i 

Materials  strength  of. 240 

Metals  and  alloys  other  than  iron  and 

steel 642 

Modulus  of  elasticity 240 

approximate  values 245 

Modulus  of  rupture,  for  beams 293 

Moeller,  cast-iron  columns.  . .  .  373 

Moment  of  deviation 108 

Mi  >ment  of  inertia 106 

Moments  of  inertia  about  different  axes.  113 

components  of 117 

Moments  of  inertia,  equal  values  of 1 16 

examples 1 18 

of  Phoenix  columns 286 

of  plane  figure?  about  parallel  axes.  .  no 

of  plane  surface 107 

of  sections 275 

of  solids  around  parallel  axes 117 

polar,  of  plane  figures in 

principal 114 

Momentum 1 1 

Mortar  (cement) 120 

Motion  and  rest 2 

Motion,  Newton's  first  law  of 4,  9 

Newton's  second  law  of 13 

01.  curved  lin* 95 

on  inclined  plane  ., 92 

relativity  of i 

under  influence  of  gravity 87 

•uniform 76 

uniformly  v.  r--in.'* 76 


Motion,  uniformly  varying  rectilinear 

Motions,  parallelogram  of 

polygon  of 

m,  value  of.  .  , 


NAILS  in  one  pound.  .  . 
Neutral  axis  of  beams. 
Notation,  Bow's.  .  , 


OAK  columns. 


PAPPUS'S  theorems 

Pendulum,  cycloidal 

simple  circular 

Pig-iron 

Plates,  flat,  strength  of 

thickness  of 

Polygonal  frame 

Projectile,  unresisted 

Punching  and  drilling  plates. 


RADIOS  of  gyration ". 

Rankine    strength  of  timber 

Reduction  in   rolls,   Beardslee,  wroxight- 

iron 

Resilience  of  a  beam 

of  tie-bar 

Resistance,  line  of 

Resistance,  line  of  maximum  and  mini- 
mum  

to  direct  compression 

to  shearing 

true  line  of 

Rest  effect.  Beardslee 

bodies,  rectilinear  transference  of 
force   

rotation  of 

statics  of 

Riveted  ioint.-> 

rules  by  P.  Schwamb 

transverse  strength 

punching  and  drilling 

Watertown-arsenal  tests 

Rodman,  strength  of   timber 

3.oofs,  estimation  of  load 

weight  of  materials 

3.oof-trusses 1 38, 

determination  of  stresses 150, 

distribution  of  load 

examples 


15 
15 

870 


269 

M3 


234 

99 

97 

35<> 

900 

908- 

MS 

89 


107 
647 

401 

305 
24? 
795 

799 
253 

256 
803 
401 

29 

105 
29 
54<> 
550 
555 
S4& 
570 
650 
163. 
151 
1  60 
165 
163 
1  6s 


2ND  EX. 


927 


Roof -trasses,  general  remarks 172 

with  loads  at  lower  joints 171 


Rope,  wire 

Rosset's  tests  of  cast-iron. 

Rupture,  modulus  for  beams.   . 


639 
363 
293 


SCHEFFLER'S  method  with  arches 805 

mode  of  correcting  joints 811 

Scissor-beam  truss 179 

without  horizontal  tie •.    180 

Seasoned  columns 655 

Seasoning,  effect  on  timber 712 

Semi-girder,  greatest  stresses 193 

Shafting,  strength  of 333,  539 

Shafts,  transverse  deflection 337 

under  combined   torsion   and    bend- 
ing     889 

Shape  of  specimen 351 

Kirkaldy 394 

Shearing-force 18 

actual  for  bridge-trusses 20 

in  beams 27 

Shearing,  resistance  to 251 

Shearing-strength  of  iron  and  steel. 4 19,  535 

Shearing  of  timber. 71 

of  timber  beams 695 

Slating,  weight 15 

Slope  of  beams 301 

Slotted  cross-head 102 

Snow,  weight  of 151 

Springs 338 

torsional  strength 339 

transverse  strength 343 

Spruce  beams 675 

Stability  of  a  buttress 793 

of  an  arch Soo 

of  position 793 

Standard  specifications  for  cast-iron.  .  .  .    385 

for  cement 7  24,  7  29 

for  steel 450 

for  wrought-iron .    395 

Steel,   composition,   kinds,   and   charac- 
teristics     445 

crystallization 498 

effects  of  temperature 506 

eye-bars 487 

tensile  strength  and  elasticity 

459,  464,  487 

torsional  strength 487 

wire 487 


PAGE 
714 

Strain 24o 

resultant 857 

Strains 852 

in  terms  of  distortions 855 

relation  to  stresses 865 

Strength,  breaking  and  working 245 

of  columns 323 

of  hooks 323 

of  materials 240 

of  materials,  general  remarks 350 

of  shafting 333 

Stress 240 

centre  of 263 

compound 861 

converse  of  ellipse 884 

ellipse 881 

graphical  representation 261 


intensity  of 

relation  to  strains 

simple 

tangential 

uniform •  •  •  • 

uniformly  varying 

Stress,    uniformly    varying,    amounting 
to  a  statical  couple 

Stresses 

Stresses,  composition  of 872 

conjugate 87  2 

equilibrium  of 865 

greatest,  in  girder 193 

in  roof-trusses,  determination  of 150 

mode  of  ascertaining,  in  a  beam 291 

parallel  to  a  plane,  composition.'.  .  .  .    875 
principal 877 


260 
865 
860 
861 
264 
265 

266 
859 


principal,  determination  of 

Stretching  and  tearing 

Strut 

Struts,  short 

Suspension-rod  of  uniform  strength.  .  . . 


878 
242 
138 
323 
250 


TEMPERATURE,  effect  on  iron  and  steel.  .    506 

Tensile  limit.  Beardslee 400 

Tension  of  timber 651 

Gheory  of  elasticity 852 

Timber  beams,   immediate   modulus   of 

elasticity 695 

beams     modulus    of    elasticity    for 

use 695 

beams,  time  tests.  .  .  .    OSS 


928 


INDEX. 


Timber  columns 


compression 

effect  of  seasoning 

factor  of  safety 

floors 

framing-joints 

general  remarks 

longitudinal    shearing    in    beams.  . 

posts,  rules  for  strength,  Ely 

strength  as  given  by  Rankine , 

strength  as  given  by  Rodman 

tension 

tests  by  Bauschinger 

transverse  strength 

Time  of  descent  down  a  curve 

Time  tests  on  timber  beams 

Torsional  strength  of  iron  and  steel 

Torsional  strength  of  springs 

Torsion  and  bending  combined 

Translation  and  rotation  combined 

Transverse  deflection  of  shafts 

Transverse  strength  of  cast-iron 

springs 

timber 

wrought-iron 

Travelling-load 

Triangular  frame 

Triangular  truss,  wind  pressure 

Truss,  Bollman's 

Fink's 

hammer-beam 

scissor-beam 

triangtilar,  wind  pressure 

Trusses,  bridge 

methods  for  determining  stresses 

roof 138, 


PAG 

•  65 
.  65 

•  7i 
672 
699 
698 
712 
695 
671 
647 
650 
651 
707 
673 

96 
688 
487 
339 
889 

44 
337 
375 
343 
673 
440 
192 
141 
147 
219 
217 
176 
179 
147 
184 
140 
1 66 


PAGE 

Trusses,  roof,  determination  of  stresses..    150 
Twisting  and  bending  combined 338 


VELOCITY. 


•: 2,  75 

WATERTOWN  ARSENAL,  steel  eye-bars.  .  .    487 

tests  of  bridge  columns 421 

tests  of  riveted  joints 546 

Weight  of  materials  for  roofs 151 

of  snow 151 

Weyrauch's  formula 255,  525 

White-pine  beams 686 

columns 659 

Wind  pressure 152 

triangular  truss 147 

Wire,  and  wire  rope 639 

experiments 526 

Working-strength 245 

f  beams 293 

Work,  mechanical 77 

tinder  oblique  force 104 

unit  of 77 

Wrought-iron   beams,   strength  and  de- 
flection     440 

Wrought-in  n  characteristics 391 

compressive  strength 413 

crystallization 498 

effect  of  temperature 506 

eye-bars 412 

tests  by  Beardslee 399 

tests  by  Kirkaldy.  .• 394 

transverse  strength 440 


FELLOW-PINE  beams,  strength  and  elas- 
ticity     68 1 

columns,  tables 653,  664 


SHORT-TITLE     CATALOGUE 

OF  THE 

PUBLICATIONS 

OP 

JOHN   WILEY   &    SONS, 

NEW  YORK, 
LONDON:  CHAPMAN  &  HALL,  LIMITED. 

ARRANGED  UNDER  SUBJECTS. 


Descriptive  circulars  sent  on  application.  Books  marked  with  an  asterisk  (*)  are  sold 
at  net  prices  only,  a  double  asterisk  (**)  books  sold  under  the  rules  of  the  American 
Publishers'  Association  at  net  prices  subject  to  an  extra  charge  for  postage.  All  books 
are  bound  in  cloth  unless  otherwise  stated. 


50 


oo 
50 

00 

50 
oo 
50 
50 
50 
50 


AGRICULTURE. 

Armsby's  Manual  of  Cattle-feeding izmo,  $i  75 

Principles  of  Animal  Nutrition 8vo,    4  oo 

Budd  and  Hansen's  American  Horticultural  Manual: 

Part  I.  Propagation,  Culture,  and  Improvement I2mo, 

Part  II.  Systematic  Pomology i2mo, 

Downing's  Fruits  and  Fruit-trees  of  America 8vo, 

Elliott's  Engineering  for  Land  Drainage i2mo, 

Practical  Farm  Drainage i2mo. 

Green's  Principles  of  American  Forestry i2mo, 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (Woll.) i2mo, 

Kemp's  Landscape  Gardening i2mo, 

Maynard's  Landscape  Gardening  as  Applied  to  Home  Decoration i2mo, 

*  McKay  and  Larsen's  Principles  and  Practice  of  Butter-making 8vo, 

Sanderson's  Insects  Injurious  to  Staple  Crops i2mo, 

Insects  Injurious  to  Garden  Crops.     (In  preparation.) 
Insects  Injuring  Fruits.     (In  preparation.) 

Stockbridge's  Rocks  and  Soils. 8vo,    2  50 

Winton's  Microscopy  of  Vegetable  Foods 8vo,    7  50 

Woll's  Handbook  for  Farmers  and  Dairymen i6mo,    i  50 

ARCHITECTURE. 

Baldwin's  Steam  Heating  for  Buildings izmo,  2  50 

Bashore's  Sanitation  of  a  Country  House I2tno,  i  oo 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  oo 

Birkmire's  Planning  and  Construction  of  American  Theatres 8vo,  3  oo 

Architectural  Iron  and  Steel 8vo,  3  50 

Compound  Riveted  Girders  as  Applied  in  Buildings 8vo,  2  oo 

Planning  and  Construction  of  High  Office  Buildings 8vo,  3  50 

Skeleton  Construction  in  Buildings 8vo,  3  oo 

Brigg's  Modern  American  School  Buildings .• 8vo,  4  oo 

Carpenter's  Heating  and  Ventilating  of  Buildings 8vo,  4  oo 

Freitag's  Architectural  Engineering 8vo,  3  50 

Fireproofing  of  Steel  Buildings 8vo,  2  50 

French  and  Ives's  Stereotomy 8vo,  2  50 

1 


Gerhard's  Guide  to  Sanitary  House-inspection i6mo,    i  oo 

Theatre  Fires  and  Panics i2mo,    i  50 

*Greene's  Structural  Mechanics 8vo,    2  50* 

Holly's  Carpenters'  and  Joiners'  Handbook i8mo,        75 

Johnson's  Statics  by  Algebraic  and  Graphic  Methods 8vo,    2  oo 

Kidder's  Architects' and  Builders' Pocket-book.  Rewritten  Edition.  i6mo,mor.,  5  oo 

Merrill's  Stones  for  Building  and  Decoration 8vo,    5  oo 

Non-metallic  Minerals :   Their  Occurrence  and  Uses 8vo,    4  oo 

Monckton's  Stair-building 4to,    4  oo 

Patton's  Practical  Treatise  on  Foundations 8vo,    5  oo 

Peabody's  Naval  Architecture 8vo,    7  50 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor.,    4  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,    3  oo 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry 8vo,    i  50 

Snow's  Principal  Species  of  Wood 8vo,    3  50 

Sondericker's  Graphic  Statics  with  Applications  to  Trusses,  Beams,  and  Arches. 

8vo,    2  oo 

Towne's  Locks  and  Builders'  Hardware i8mo,  morocco,    3  oo 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo,    6  oo 

Sheep,    6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,    5  oo 

Sheep,    5  50 

Law  of  Contracts ; 8vo,    3  oo 

Wood's  Rustless  Coatings;   Corrosion  and  Electrolysis  of  Iron  and  Steel.  .8vo,    4  oo 
Worcester  and  Atkinson's  Small  Hospitals,  Establishment  and  Maintenance, 
Su^estions  for  Hospital  Architecture,  with  Plans  for  a  Small  Hospital. 

i2mo,    i  25 
The  World's  Columbian  Exposition  of  1893 Large  4to,    i  oo 


ARMY  AND  NAVY. 

Bernadou's  Smokeless  Powder,  Nitro-cellulose,  and  the  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

*  Bruff 's  Text-book  Ordnance  and  Gunnery 8vo,  6  oo 

Chase's  Screw  Propellers  and  Marine  Propulsion 8vo,  3  oo 

Cloke's  Gunner's  Examiner 8vo,  i  50 

Craig's  Azimuth 4to,  3  50 

Crehore  and  Squier's  Polarizing  Photo-chronograph 8vo»  3  oo 

*  Davis's  Elements  of  Law 8vo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States.  .• 8vo,  7  oo 

Sheep,  7  50 

De  Brack's  Cavalry  Outposts  Duties.     (Carr.) 241110,  morocco,  2  oo 

Dietz's  Soldier's  First  Aid  Handbook i6mo,  morocco,  i  25 

*  Dredge's  Modern  French  Artillery 4to,  half  morocco,  15  oo 

Durand's  Resistance  and  Propulsion  of  Ships 8vo,  5  oo 

*  Dyer's  Handbook  of  Light  Artillery I2mo,  3  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

*  Fiebcger's  Text-book  on  Field  Fortification Small  8vo,  2  oo 

Hamilton's  The  Gunner's  Catechism i8mo,  i  oo 

*  Hoff's  Elementary  Naval  Tactics 8vo,  i  50 

Ingalls's  Handbook  of  Problems  in  Direct  Fire , 8vo,  4  oo 

*  Ballistic  Tables 8vo,  i  50 

*  Lyons's  Treatise  on  Electromagnetic  Phenomena.  Vols.  I.  and  II.  .8vo,  each,  6  oo 

*  Mahan's  Permanent  Fortifications.    (Mercur.) 8vo,  half  morocco,  7  50 

Manual  for  Courts-martial i6mo,  morocco,  i  50 

*  Mercur's  Attack  of  Fortified  Places i2mo,  2  oo 

*  Elements  of  the  Art  of  War 8vo,  4  oo 


Mctcalr's  Cost  of  Manufactures — And  the  Administration  of  Workshops.  .8vo,  5  oo 

*  Ordnance  and  Gunnery.     2  vols i2mo,  5  oo 

Murray's  Infantry  Drill  Regulations i8mo,  paper,  10 

Nixon's  Adjutants'  Manual 24010,  i  oo 

Peabody's  Naval  Architecture 8vo,  7  50 

*  Phelps's  Practical  Marina  Surveying 8vo,  2  50 

Powell's  Army  Officer's  Examiner 121110,  4  oo 

Sharpe's  Art  of  Subsisting  Armies  in  War i8mo,  morocco  i  50 

*  Tupes  and  Poole's  Manual  of  Bayonet  Exercises  and    Musketry  Fencing. 

241110,  leather,  50 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

*  Wheeler's  Siege  Operations  and  Military  Mining 8vo,  2  oo 

Winthrop's  Abridgment  of  Military  Law 12 mo,  2  50 

Woodhull's  Notes  on  Military  Hygiene i6mo,  i  50 

Young's  Simple  Elements  of  Navigation i6mo,  morocco,  2  o 

o 

ASSAYING. 

Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

i2mo,  morocco,  i  50 

Furman's  Manual  of  Practical  Assaying 8vo,  3  oo 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments.  .  .  .8vo,  3  oo 

Low's  Technical  Methods  of  Ore  Analysis 8vo,  3  oo 

Miller's  Manual  of  Assaying i2mo,  i  oo 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.) i2mo,  2  50 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  oo 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo,  4  oo 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

Wilson's  Cyanide  Processes. i2mo,  i  50 

Chlorination  Process i2mo,  i  50 

ASTRONOMY. 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

Craig's  Azimuth 4to,  3  50 

Doolittle's  Treatise  on  Practical  Astronomy 8vo,  4  oo 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo,  3  oo 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  2  50 

*  Michie  and  Harlow's  Practical  Astronomy 8vo,  3  oo 

*  White's  Elements  of  Theoretical  and  Descriptive  Astronomy i2mo,  2  oo 

BOTANY. 

Davenport's  Statistical  Methods,  with  Special  Reference  to  Biological  Variation. 

i6mo,  morocco,  i  25 

Thomd  and  Bennett's  Structural  and  Physiological  Botany i6mo,  2  25 

Westermaier's  Compendium  of  General  Botany.     (Schneider.).  ......... .8vo,  2  oo 

CHEMISTRY. 

Adriance's  Laboratory  Calculations  and  Specific  Gravity  Tables 12010,  i  23 

Allen's  Tables  for  Iron  Analysis 8vo,  3  oo 

Arnold's  Compendium  of  Chemistry.     (Mandel.) Small  8vo,  3  50 

Austen's  Notes  for  Chemical  Students i2mo,  i  50 

Bernadou's  Smokeless  Powder.— Nitro-cellulose,  and  Theory  of  the  Cellulose 

Molecule i2mo,  2  50 

*  Browning's  Introduction  to  the  Rarer  Elements 8vo,  i  50 

3 


Brush  and  Penfield's  Manual  of  Determinative  Mineralogy ivo,  4  oo 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.    (Eoltwccd.).  .8vo,  3  co 

Cohn's  Indicators  and  Test-papers i2mo,  2  oo 

Tests  and  Reagents : .  .  .  .  8vo,  3  oo 

Crafts's  Short  Course  in  Qualitative  Chemical  Analysis.   (Schaeffer.).  .  .I2mo,  i  50 
Dolezalek's  Theory  of  the   Lead  Accumulator   (Storage   Battery).        (Von 

Ende.) i2mo,  2  50 

Drechsel's  Chemical  Reactions.     (Merrill.) i2mo,  i  25 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo,  4  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Effront's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  oo 

Erdmann's  Introduction  to  Chemical  Preparations.     (Dunlap.) i2mo,  i  25 

Fletcher's  Practical  Instructions  in  Quantitative  Assaying  with  the  Blowpipe. 

lamo,  morocco,  i  50 

Fowler's  Sewage  Works  Analyses i2mc,  2  GO 

Fresenius's  Manual  of  Qualitative  Chemical  Analysis.     (Wells.) 8vo,  5  oo 

Manual  of  Qualitative  Chemical  Analysis.  Fart  I.  Descriptive.  (Wells.)  8vo,  3  oo 
System   of    Instruction    in    Quantitative    Chemical    Analysis.      (Cohn.) 

2  vols 8vo,  12  50 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Furman's  Manual  of  Practical  Assaying 8vo,  3  oo 

*  Getman's  Exercises  in  Physical  Chemistry.  .  .  .' i2mo,  2  oo 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i  25 

Grotenfelt's  Principles  of  Modern  Dairy  Practice.     (Woll.) i2rr.o,  2  oo 

Hammarsten's  Text-book  of  Physiological  Chemistry.     (Mandel.) 8vo,  4  oo 

Helm's  Principles  of  Mathematical  Chemistry.     (Morgan.) i2mo,  i  50 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Hind's  Inorganic  Chemistry 8vo,  3  oo 

*  Laboratory  Manual  for  Students I2mo,  i  oo 

Holleman's  Text-book  of  Inorganic  Chemistry.     (Cooper.) 8vo,  2  50 

Text-book  of  Organic  Chemistry.     (Walker  and  Mott.) 8vo,  2  50 

*  Laboratory  Manual  of  Organic  Chemistry.     (Walker.) i2mo,  i  oo 

Hopkins's  Oil-chemists'  Handbook 8vo,  3  oo 

Jackson's  Directions  for  Laboratory  Work  in  Physiological  Chemistry.  .8vo,  i  25 

Keep's  Cast  Iron 8vo,  2  50 

Ladd's  Manual  of  Quantitative  Chemical  Analysis i2mo,  i  oo 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,  3  oo 

*  Langworthy  and  Austen.        The   Occurrence   of  Aluminium  in  Vegetable 

Products,  Animal  Products,  and  Natural  Waters 8vo,  2  oo 

Lassar-Cohn's  Practical  Urinary  Analysis.  (Lorenz.) i2mo,  i  oo 

Application  of  Some  General  Reactions  to  Investigations  in  Organic 

Chemistry.  (Tingle.) i2irc,  i  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,  7  50 

Lob's  Electrochemistry  of  Organic  Compounds.  (Lorenz.) 8vo,  3  oo 

Lodge's  Notes  on  Assaying  and  Metallurgical  Laboratory  Experiments.  ..  .8vo,  3  oo 

Low's  Technical  Method  of  Ore  Analysis 8vo,  3  oo 

Lunge's  Techno-chemical  Analysis.  (Cohn.) i2mo  i  oo 

*  McKay  and  Larsen's  Principles  and  Practice  of  Butter-making 8vo.  i  50 

Mandel's  Handbook  for  Bio-chemical  Laboratory i2rro,  i  50 

*  Martin's  Laboratory  Guide  to  Qualitative  Analysis  with  the  Blowpipe .  .  I2rro,  60 
Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

3d  Edition,  Rewritten 8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) 1211:0,  i  25 

Matthew's  The  Textile  Fibres 8vo,  3  50 

Meyer's  Determination  of  Radicles  in  Carbon  Compounds.     (Tingle.).  .i2mo,  i  oo 

Miller's  Manual  of  Assaying I2mo,  i  oo 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.) .  .  .  .  i2mo,  2  50 

Mixter's  Elementary  Text-book  of  Chemistry i2mo,  i  50 

Morgan's  An  Outline  of  the  Theory  of  Solutions  and  its  Results I2mo,  i  oo 

4 


Morgan's  Elements  of  Physical  Chemistry I2mo,  3  oo 

*  Physical  Chemistry  for  Electrical  Engineers 12010,  i  50 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

Mulliken's  General  Method  for  the  Identification  of  Pure  Organic  Compounds. 

Vol.  I Large  8vo,  5  oo 

O'Brine's  Laboratory  Guide  in  Chemical  Analysis 8vo,  2  oo 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Ostwald's  Conversations  on  Chemistry.     Part  One.     (Ramsey.) i2mo,  i  50 

Part  Two.     (Turnbull.) i2mo,  200 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mir.eral  Tests. 

8vo,  paper,  50 

Pictet's  The  Alkaloids  and  their  Chemical  Constitution.     (Biddle.) 8vo,  5  oo 

Pinner's  Introduction  to  Organic  Chemistry.     (Austen.) i2mo,  i  50 

Poole's  Calorific  Power  of  Fuels 8vo,  3  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo,  i  25 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Richards  and  Woodman's   Air,  Water,  and    Food  from  a  Sanitary  Stand- 
point   8vo,  2  oa 

Ricketts  and  Russell's  Skeleton  Notes  upon  Inorganic  Chemistry.     (Part  I. 

Non-metallic  Elements.) 8vo,  morocco,  75 

Ricketts  and  Miller's  Notes  on  Assaying 8vo,  3  oo« 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  3  50- 

Disinfection  and  the  Preservation  of  Food 8vo,  4  oo 

Riggs's  Elementary  Manual  for  the  Chemical  Laboratory 8vo,  i  25 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo,  4  oo> 

Rostoski's  Serum  Diagnosis.     (Bolduan.) i2mo,  i  oo> 

Ruddiman's  Incompatibilities  in  Prescriptions 8vo,  2  oo 

*  Whys  in  Pharmacy i2mo,  i  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Salkowski's  Physiological  and  Pathological  Chemistry.     (Orndorff.) 8vo,  2  50 

Schimpf's  Text-book  of  Volumetric  Analysis i2mo,  2  50- 

Essentials  of  Volumetric  Analysis i2mo,  i  25 

*  Qualitative  Chemical  Analysis 8vo,  i  25 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  oo 

Handbook  for  Cane  Sugar  Manufacturers i6mo,  morocco,  3  oo 

Stockbridge's  Rocks  and  Soils .. 8vo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat 8vo,  i  50 

*  Descriptive  General  Chemistry 8vo,  3  oo 

Treadwell's  Qualitative  Analysis.     (Hall.) 8vo,  3  oo 

Quantitative  Analysis.     (Hall.) 8vo,  4  oo 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Van  Deventer's  Physical  Chemistry  for  Beginners.     (Boltwood.) I2mo,  i  50 

*  Wai'ke's  Lectures  on  Explosives 8vo,  4  oo 

Ware's  Beet-sugar  Manufacture  and  Refining Small  8vo,  cloth,  4  oo 

Washington's  Manual  of  the  Chemical  Analysis  of  Rocks 8vo,  2  oo 

Wassermann's  Immune  Sera:  Haemolysins,  Cytotoxins,  and  Precipitins.    (Bol- 
duan.)   i2mo,  i  oo 

Wells's  Laboratory  Guide  in  Qualitative  Chemical  Analysis 8vo,  i  50 

Short  Course  in  Inorganic  Qualitative  Chemical  Analysis  for  Engineering 

Students I2mo,  i  50 

Text-book  of  Chemical  Arithmetic lamo,  i  25 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Wilson's  Cyanide  Processes I2mo,  i  50 

Chloriration  Process i2mo,  i  5° 

Winton's  Microscopy  of  Vegetable  Foods ,  .  .8vo,  7  5<> 

Wulling's    Elementary    Course    in  Inorganic,  Pharmaceutical,  and  Medical 

Chemistry I2mo,  2  oo- 

5 


CIVIL  ENGINEERING. 

BRIDGES    AND    ROOFS        HYDRAULICS.       MATERIALS   OF    ENGINEERING. 
RAILWAY  ENGINEERING. 

Baker's  Engineers'  Surveying  Instruments 12010,  3  oo 

Bixby's  Graphical  Computing  Table Paper  19^X24!  inches.  25 

**  Burr's  Ancient  and  Modern  Engineering  and  the  Isthmian  Cana  *.     (Postage, 

27  cents  additional.) 8vo,  3  50 

Comstock's  Field  Astronomy  for  Engineers 8vo,  2  50 

Davis's  Elevation  and  Stadia  Tables 8vo,  i  oo 

Elliott's  Engineering  for  Land  Drainage 12010,  i  50 

Practical  Farm  Drainage I2mo,  i  oo 

*Fiebeger's  Treatise  on  Civil  Engineering 8vo,  5  oe 

Flemer's  Phototopographic  Methods  and  Instruments 8vo,  5  oo 

Folwell's  Sewerage.     (Designing  and  Maintenance.) 8vo,  3  oo 

Freitag's  Architectural  Engineering.     2d  Edition,  Rewritten 8vo,  3  50 

French  and  Ives's  Stereotomy 8vo,  2  50 

Goodhue's  Municipal  Improvements, i2mo,  i  75 

Goodrich's  Economic  Disposal  of  Towns'  Refuse 8vo,  3  50 

Gore's  Elements  of  Geodesy 8vo,  2  50 

Hayford's  Text-book  of  Geodetic  Astronomy 8vo,  3  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Howe's  Retaining  Walls  for  Earth i2mo,  i  25 

Johnson's  (J.  B.)  Theory  and  Practice  of  Surveying Small  8vo,  4  oo 

Johnson's  (L.  J.)  Statics  by  Algebraic  and  Graphic  Methods 8vo,  2  oo 

Laplace's  Philosophical  Essay  on  Probabilities.    (Truscott  and  Emory.) .  i2mo,  2  oo 

Mahan's  Treatise  on  Civil  Engineering.     (1873.)     (Wood.) 8vo,  5  oo 

*  Descriptive  Geometry 8vo,  i  50 

Merriman's  Elements  of  Precise  Surveying  and  Geodesy 8vo,  2  50 

Merriman  and  Brooks's  Handbook  for  Surveyors i6mo,  morocco,  2  oo 

Nugent's  Plane  Surveying 8vo,  3  50 

Ogden's  Sewer  Design.  . i2mo,  2  oo 

Patton's  Treatise  on  Civil  Engineering 8vo  half  leather,  7  50 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

Rideal's  Sewage  and  the  Bacterial  Purification  of  Sewage 8vo,  3  50 

Siebert  and  Biggin's  Modern  Stone-cutting  and  Masonry 8vo,  i  50 

Smith's  Manual  of  Topographical  Drawing.     (McMillan."* 8vo,  2  50 

Sondericker's  Graphic  Statics,  with  Applications  to  Trusses,  Beams,  and  Arches. 

8vo,  2  co 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

*  Trautwire's  Civil  Engineer's  Focket-book i6mo,  morocco,  5  oo 

Wait's  Engineering  and  Archi'ectural  Jurisprudence 8vo,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo,  5  oo 

Sheep,  5  50 

Law  ©f  Contracts 8vo,  3  oo 

Warren's  Stereotomy — Problems  in  Stone-cutting 8vo,  2  50 

Webb's  Problems  in  the  Use  and  Adjustment  of  Engineering  Instruments. 

i6mo,  morocco,  i  25 

Wilson's  Topographic  Surveying 8vo,  3  So 


BRIDGES  /ND  ROOFS. 

Boiler's  Practical  Treatise  on  the  Construction  of  Iron  Highway  Bridges.  .8vo,  2  oo 

*       Thames  River  Bridge 4*0,  paper,  5  oo 

Burr's  Course  on  the  Stresses  in  Bridges  and  Roof  Trusses,  Arched  Ribs,  and 

Suspension  Bridges 8vo,  3  50 

6 


Burr  and  Falk's  Influence  Lines  for  Bridge  and  Roof  Computations.  .  .  .8vo,  3  oo 

Design  and  Construction  of  Metallic  Bridges 8vo,  5  oo 

Du  Bois's  Mechanics  of  Engineering.     Vol.  II Small  4to,  10  oo 

Foster's  Treatise  on  Wooden  Trestle  Bridges 4to,  5  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

Greene's  Roof  Trusses 8vo,  i  25 

Budge  Trusses 8vo,  2  50 

Arches  in  Wood,  Iron,  and  Stone 8vo,  2  50 

Howe's  Treatise  on  Arches * 8vo,  4  oo 

Design  of  Simple  Roof-trusses  in  Wood  and  Steel 8vo,  2  oo 

Johnson,  Bryan,  and  Turneaure's  Theory  and  Practice  in  the  Designing  of 

Modern  Framed  Structures Small  4to,  10  oo 

Merriman  and  Jacoby's  Text-book  on  Roofs  and  Bridges: 

Part  I.     Stresses  in  Simple  Trusses 8vo,  2  50 

Part  II.     Graphic  Statics 8vo,  2  50 

Part  III.     Bridge  Design 8vo,  2  50 

Part  IV.     Higher  Structures 8vo,  2  50 

Morison's  Memphis  Bridge 4to,  10  oo 

Waddell's  De  Pontibus,  a  Pocket-book  for  Bridge  Engineers.  .  i6mo,  morocco,  2  oo 

*Specifications  for  Steel  Bridges i2mo,  50 

Wright's  Designing  of  Draw-spans.     Two  parts  in  one  volume 8vo,  3  50 


HYDRAULICS. 

Bazin's  Experiments  upon  the  Contraction  of  the  Liquid  Vein  Issuing  from 

an  Orifice.     (Trautwine.) 8vo,  2  oo 

Bovey's  Treatise  on  Hydraulics 8vo,  5  oo 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Diagrams  of  Mean  Velocity  of  Water  in  Open  Channels paper,  i  50 

Hydraulic  Motors 8vo,  2  oo 

Coffin's  Graphical  Solution  of  Hydraulic  Problems i6mo,  morocco,  2  50 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  oo 

Folwell's  Water-supply  Engineering 8vo,  4  oo 

Frizell's  Water-power 8vo,  5  oo 

Fuertes's  Water  and  Public  Health i2mo,  i  50 

Water-filtration  Works i2mo,  2  50 

Ganguillet  and  Kutter's  General  Formula  for  the  Uniform  Flow  of  Water  in 

Rivers  and  Other  Channels.     (Bering  and  Trautwine.) 8vo,  4  oo 

Hazen's  Filtration  of  Public  Water-supply 8vo,  3  oo 

Hazlehurst's  Towers  and  Tanks  for  Water-works 8vo,  2  50 

Herschel's  115  Experiments  on  the  Carrying  Capacity  of  Large,  Riveted,  Metal 

Conduits 8vo,  2  oo 

Mason's  Water-supply.     (Considered  Principally  from  a  Sanitary  Standpoint.) 

8vo,  4  oo 

Merriman's  Treatise  on  Hydraulics 8vo,  5  oo, 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  oo 

Schuyler's   Reservoirs  for  Irrigation,   Water-power,  and   Domestic   Water- 
supply Large  8vo,  5  oo 

**  Thomas  and  Watt's  Improvement  of  Rivers.     (Post.,  440.  additional.). 410,  6  oo 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Wegmann's  Design  and  Construction  of  Dams 4to,  5  oo 

Water-supply  of  the  City  of  New  York  from  1658  to  1895 410,  10  oo 

Williams  and  Hazen's  Hydraulic  Table's 8vo,  i  50 

Wilson's  Irrigation  Engineering Small  8vo,  4  oo 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Turbines 8vo,  2  50 

Elements  of  Analytical  Mechanics 8vo,  3  oo 

7 


MATERIALS  OF  ENGINEERING. 

Baker's  Treatise  on  Masonry  Construction 8vo,  5  oo 

Roads  and  Pavements 8vo,  5  oo 

Black's  United  States  Public  Works Oblong  4to,  5  oo 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering 8vo,  7  50 

Byrne's  Highway  Construction 8vo,  5  oo 

Inspection  of  the  Materials  and  Workmanship  Employed  in  Construction. 

i6mo,  3  oo 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Du  Bois's  Mechanics  of  Engineering.     Vol.  I Small  4to,  7  50 

*Eckel's  Cements,  Limes,  and  Plasters 8vo,  6  oo 

Johnson's  Materials  of  Construction Large  8vo,  6  oo 

Fowler's  Ordinary  Foundations 8vo,  3  50 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Marten's  Handbook  on  Testing  Materials.     (Henning.)     2  vols 8vo,  7  50 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merrill's  Stones  for  Building  and  Decoration 8vo,  5  oo 

Merriman's  Mechanics  of  Materials 8vo,  5  oo 

Strength  of  Materials i2mo,  i  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Patton's  Practical  Treatise  on  Foundations 8vo,  5  oo 

Richardson's  Modern  Asphalt  Pavements 8vo,  3  oo 

Richey's  Handbook  for  Superintendents  of  Construction i6mo,  mor.,  4  oo 

Rockwell's  Roads  and  Pavements  in  France i2mo,  i  25 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Materials  of.  Machines i2mo,  i  oo 

Snow's  Principal  Species  of  Wood 8vo,  3  50 

Spalding's  Hydraulic  Cement i2mo,  2  oo 

Text-book  on  Roads  and  Pavements i2nao,  2  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced 8vo,  5  oo 

Thurston's  Materials  of  Engineering.     3  Parts 8vo,  8  oo 

Part  I.     Non-metallic  Materials  of  Engineering  and  Metallurgy 8vo,  2  00 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Thurston's  Text-book  of  the  Materials  of  Construction 8vo,  5  oo 

Tillson's  Street  Pavements  and  Paving  Materials 8vo,  4  oo 

Waddell's  De  Pontibus.    (A  Pocket-book  for  Bridge  Engineers.).  .i6mo,  mor.,  2  oo 

Specifications  for  Steel  Bridges i2mo,  i  25 

Wood's  (De  V.)  Treatise  om  the  Resistance  of  Materials,  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,  2  oo 

Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

Steel 8vo,  4  oo 


RAILWAY  ENGINEERING. 

Andrew's  Handbook  for  Street  Railway  Engineers 3x5  inches,  morocco,  I  25 

Berg's  Buildings  and  Structures  of  American  Railroads 4to,  5  oo 

Brook's  Handbook  of  Street  Railroad  Location i6mo,  morocco,  i  50 

Butt's  Civil  Engineer's  Field-book i6mo,  morocco,  2  50 

Crandall's  Transition  Curve i6mo,  morocco,  i  50 

Railway  and  Other  Earthwork  Tables 8vo,  I  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  .  i6mo,  morocco,  5  oo. 


Dredge's  History  of  the  Pennsylvania  Railroad:    (1879) Paper,  5  oo 

*  Drinker's  Tunnelling,  Explosive  Compounds,  and  Rock  Drills. 4to,  half  mor.,  25  oo 

Fisher's  Table  of  Cubic  Yards Cardboard,  25 

Godwin's  Railroad  Engineers'  Field-book  and  Explorers'  Guide.  .  .  i6mo,  mor.,  2  50 

Howard's  Transition  Curve  Field-book i6mo,  morocco,  i  50 

Hudson's  Tables  for  Calculating  the  Cubic  Contents  of  Excavations  and  Em- 
bankments  8vo,  i  oo 

Molitor  and  Beard's  Manual  for  Resident  Engineers i6mo,  i  oo 

Nagle's  Field  Manual  for  Railroad  Engineers i6mo,  morocco,  3  oo 

Philbrick's  Field  Manual  for  Engineers i6mo,  morocco,  3  oo 

Searles's  Field  Engineering i6mo,  morocco,  3  oo 

Railroad  Spiral i6mo,  merocco,  i  50 

Taylor's  Prismoidal  Formulae  and  Earthwork 8vo,  i  50 

*  Trautwine's  Method  of  Calculating  the  Cube  Contents  of  Excavations  and 

Embankments  by  the  Aid  of  Diagrams 8vo,  2  oo 

The  Field  Practice  of  Laying  Out  Circular  Curves  for  Railroads. 

i2mo,  morocco,  2  50 

Cross-section  Sheet Paper,  25 

Webb's  Railroad  Construction i6mo,  morocco,  5  oo 

Wellington's  Economic  Theory  of  the  Location  of  Railways Small  8vo,  5  oo 


DRAWING. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  oo 

*  "  "  "        Abridged  Ed 8vo,  i  50 

Coolidge's  Manual  of  Drawing 8vo,  paper  i  oo 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  Engi- 
neers  Oblong  4to,  2  50 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo,  2  50 

Hill's  Text-book  on  Shades  and  Shadows,  and  Perspective 8vo,  2  oo 

Jamison's  Elements  of  Mechanical  Drawing 8vo,  2  50 

Advanced  Mechanical  Drawing 8vo,  2  oo 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

MacCord's  Elements  of  Descriptive  Geometry 8vo,  3  oo 

Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Mechanical  Drawing 4to,  4  co 

Velocity  Diagrams 8vo,  i  50 

MacLeod's  Descriptive  Geometry Small  8vo,  i  50 

*  Mahan's  Descriptive  Geometry  and  Stone-cutting 8vo,  i  50 

Industrial  Drawing.  (Thompson.) 8vo,  3  50 

Moyer's  Descriptive  Geometry 8vo,  2  oo 

Reed's  Topographical  Drawing  and  Sketching 4to,  5  oo 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Robinson's  Principles  of  Mechanism 8vo,  3  09 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Smith's  (R.  S.)  Manual  of  Topographical  Drawing.  (McMillan.) 8vo,  2  50 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo,  3  oo 

Warren's  Elements  of  Plane  and  Solid  Free-hand  Geometrical  Drawing.  12 mo, 


Drafting  Instruments  and  Operations i2mo, 

Manual  of  Elementary  Projection  Drawing i2mo, 

Manual  of  Elementary  Problems  in  the.  Linear  Perspective  of  Ferm  and 

Shadow "...'.  1 2 mo, 

Plane  Problems  in  Elementary  Geometry i2mo, 

9 


Warren's  Primary  Geometry izmo,  75 

Elements  of  Descriptive  Geometry,  Shadows,  and  Perspective 8vo,  3  50 

General  Problems  of  Shades  and  Shadows 8vo,  3  oo 

Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Problems,  Theorems,  and  Examples  in  Descriptive  Geometry 8vo,  2  50 

Weisbach's    Kinematics  [and    Power    of    Transmission.        (Hermann    and 

Klein.) 8vo,  5  oo 

Whelpley's  Practical  Instruction  in  the  Art  of  Letter  Engraving 12 mo,  2  oo 

Wilson's  (H.  M.)  Topographic  Surveying 8vo,  3  50 

Wilson's  (V.  T.)  Free-hand  Perspective 8vo,  2  50 

Wilson's  (V.  T.)  Free-hand  Lettering 8vo,  I  oo 

Woolf's  Elementary  Course  in  Descriptive  Geometry Large  8vo,  3  oo 

ELECTRICITY  AND  PHYSICS. 

Anthony  and  Brackett's  Text-book  of  Physics.     (Magie.) Small  8vo,  3  oo 

Anthony's  Lecture-notes  on  the  Theory  of  Electrical  Measurements.  .  .  .  I2mo,  i  oo 

Benjamin's  History  of  Electricity 8vo,  3  oo 

Voltaic  Cell 8vo,  3  oo 

Classen's  Quantitative  Chemical  Analysis  by  Electrolysis.     (Boltwood.).8vo,  3  oo 

Crehore  and  Squier's  Polarizing  Photo-chronograph 8vo,  3  oo 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  i6mo,  morocco,  5  oo 
Dolezalek's    Theory   of    the    Lead   Accumulator    (Storage    Battery).      (Von 

Ende.).  .  .  . i2mo,  2  50 

Duhem's  Thermodynamics  and  Chemistry.     (Burgess.) 8vo,  4  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power 12 mo,  3  oo 

Gilbert's  De  Magnete.     (Mottelay.) 8vo,  2  50 

Hanchett's  Alternating  Currents  Explained I2mo,  i  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

Holman's  Precision  of  Measurements 8vo,  2  oo 

Telescopic   Mirror-scale  Method,  Adjustments,  and  Tests.  . .  .Large  8vo,  75 

Kinzbrunner's  Testing  of  Continuous-current  Machines 8vo,  2  oo 

Landauer's  Spectrum  Analysis.     (Tingle.) 8vo,  3  oo 

Le  Chateliers  High-temperature  Measurements.  (Boudouard — Burgess.)  i2mo,  3  oo 

Lob's  Electrochemistry  of  Organic  Compounds.     (Lorenz.) 8vo,  3  oo 

*  Lyons'?  Treatise  on  Electromagnetic  Phenomena.   Vols.  I.  and  II.  8vo,  each,  6  oo 

*  Michie's  Elements  of  Wave  Motion  Relating  to  Sound  and  Light 8vo,  4  oo 

Niaudet's  Elementary  Treatise  on  Electric  Batteries.     (Fishback.) i2mo,  2  50 

*  Rosenberg's  Electrical  Engineering.     (Haldane  Gee — Kinzbrunner.).  .  .8vo,  i  50 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  1 8vo,  2  50 

Thurston's  Stationary  Steam-engines 8vo,  2  50 

*  Tillman's  Elementary  Lessons  in  Heat 8vo,  i   50 

Tory  and  Pitcher's  Manual  of  Laboratory  Physics Small  8vo,  2  oo 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 

LAW. 

*  Davis's  Elements  of  Law 8vo,  2  50 

*  Treatise  on  the  Military  Law  of  United  States 8vo,  7  oo 

*  Sheep,  7  50 

Manual  for  Courts-martial i6mo,  morocco,  i  50 

Wait's  Engineering  and  Architectural  Jurisprudence 8vo ,  6  oo 

Sheep,  6  50 

Law  of  Operations  Preliminary  to  Construction  in  Engineering  and  Archi- 
tecture  8vo  5  oo 

Sheep,  5  SO 

Law  of  Contracts 8vo,  3  oo 

Winthrop's  Abridgment  of  Military  Law I2mo>  2  So 

10 


MANUFACTURES. 

Bernadou's  Smokeless  Powder — Nitro-cellulose  and  Theory  of  the  Cellulose 

Molecule 1 2mo,  2  50 

Bolland's  Iron  Founder i2mof  2  50 

"  The  Iron  Founder,"  Supplement i2mo,  2  50 

Encyclopedia  of  Founding  and  Dictionary  of  Foundry  Terms  Used  in  the 

Practice  of  Moulding i2mo,  3  oo 

*  Eckel's  Cements,  Limes,  and  Plasters 8vo,  6  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

Effront's  Enzymes  and  their  Applications.     (Prescott.) 8vo,  3  oo 

Fitzgerald's  Boston  Machinist izmo,  i  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  oo 

Hopkin's  Oil-chemists'  Handbook 8vo,  3  oo 

Keep's  Cast  Iron 8vo,  2  50 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control Large  8vo,  7  50 

*  McKay  and  Larsen's  Principles  and  Practice  of  Butter-making 8vo,  i  50 

Matthews's  The  Textile  Fibres 8vo,  3  50 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Metcalfe's  Cost  of  Manufactures — And  the  Administration  of  Workshops. 8vo,  5  oo 

Meyer's  Modern  Locomotive  Construction 4to,  10  oo 

Morse's  Calculations  used  in  Cane-sugar  Factories i6mo,  morocco,  i  50 

*  Reisig's  Guide  to  Piece-dyeing 8vo,  25  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Press-working  of  Metals 8ro,  3  oo 

Spalding's  Hydraulic  Cement i2mo,  2  oo 

Spencer's  Handbook  for  Chemists  of  Beet-sugar  Houses i6mo,  morocco,  3  oo 

Handbook  for  Cane  Sugar  Manufacturers i6mo,  morocco,  3  oo 

Taylor  and  Thompson's  Treatise  on  Concrete,  Plain  and  Reinforced.  .  .  .  .8vo,  5  oo 
Thurston's  Manual  of  Steam-boilers,  their  Designs,  Construction  and  Opera- 
tion  8vo,  5  oo 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Ware's  Beet-sugar  Manufacture  and  Refining Small  8vo,  4  oo 

West's  American  Foundry  Practice , i2mo,  2  50 

Moulder's  Text-book i2mo,  2  50 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Rustless  Coatings:   Corrosion  and  Electrolysis  of  Iron  and  Steel.  ,8vo,  4  oo 

MATHEMATICS. 

Baker's  Elliptic  Functions 8vo,  i  50 

*  Bass's  Elements  of  Differential  Calculus 121110,  4  oo 

Briggs's  Elements  of  Plane  Analytic  Geometry i2mo, 

Compton's  Manual  of  Logarithmic  Computations 1 2ino, 

Davis's  Introduction  to  the  Logic  of  Algebra 8vo, 

*  Dickson's  College  Algebra Large  i2mo, 

*  Introduction  to  the  Theory  of  Algebraic  Equations Large  i2mo, 

Emch's  Introduction  to  Projective  Geometry  and  its  Applications 8vo, 

Halsted's  Elements  of  Geometry 8vo, 

Elementary  Synthetic  Geometry 8vo, 


oo 
50 
50 
50 
35 
50 
75 
50 
Rational  Geometry i2mo,  75 

*  Johnson's  (J.  B.)  Three-place  Logarithmic  Tables:   Vest-pocket  size. paper,        15 

100  copies  for    5  oo 

*  Mounted  on  heavy  cardboard,  8  X  TO  inches,        25 

10  copies  for    2  oo 
Johnson's  (W.  W.)  Elementary  Treatise  on  Differential  Calculus   .Small  8vo,    3  oo 

Elementary  Treatise  on  the  Integral  Calculus Small  8vo,    I  50 

11 


Johnson's  (W.  W.)  Curve  Tracing  in  Cartesian  Co-ordinates iimo,     i  oo 

Johnson's  (W.  W.)  Treatise  on  Ordinary  and  Partial  Differential  Equations. 

Small  8vo,    3  50 
Johnson's  (W.  W.)  Theory  of  Errors  and  the  Method  of  Least  Squares.  i2mo,     i  50 

*  Johnson's  (W.  W.)  Theoretical  Mechanics i2mo,    3  oo 

Laplace's  Philosophical  Essay  on  Probabilities.    (Truscott  and  Emory.).  i2mo,    2  oo 

*  Ludlow  and  Bass.     Elements  of  Trigomometry  and  Logarithmic  and  Other 

Tables 8vo,    3  oo 

Trigonometry  and  Tables  published  separately Each,    2  oo 

*  Ludlow's  Logarithmic  and  Trigonometric  Tables 8vo,     i  oo 

Mathematical  Monographs.     Edited  by  Mansfield  Merriman  and  Robert 

S.  Woodward Octavo,  each     i  oo 

No.  i.  History  of  Modern  Mathematics,  by  David  Eugene  Smith. 
No.  2.  Synthetic  Projective  Geometry,  by  George  Bruce  Halsted. 
No.  3.  Determinants,  by  Laenas  Gifford  Weld.  No.  4.  Hyper- 
bolic Functions,  by  James  McMahon.  No.  5.  Harmonic  Func- 
tions, by  William  E.  Byerly.  No.  6.  Grassmann's  Space  Analysis, 
by  Edward  W.  Hyde.  No.  7.  Probability  and  Theory  of  Errors, 
by  Robert  S.  Woodward.  No.  8.  Vector  Analysis  and  Quaternions, 
by  Alexander  Macfarlane.  No.  9.  Differential  Equations,  by 
William  Woolsey  Johnson.  No.  10.  The  Solution  of  Equations, 
by] Mansfield  Mernman.  No.  n.  Functions  of  a  Complex  Variable, 
by  Thomas  S.  Fiske. 

Maurer's  Technical  Mechanics 8vo,    4  oo 

Merriman's  Method  of  Least  Squares 8vo,    2  oo 

Rice  and  Johnson's  Elementary  Treatise  on  the  Differential  Calculus. .  Sm.  8vo,    3  oo 

Differential  and  Integral  Calculus.     2  vols.  in  one Small  8vo,    2  50 

Wood's  Elements  of  Co-ordinate  Geometry 8vo,    2  oo 

Trigonometry:   Analytical,  Plane,  and  Spherical i2mo,    1  oo 


MECHANICAL  ENGINEERING. 
MATERIALS  OF  ENGINEERING,  STEAM-ENGINES  AND  BOILERS. 

Bacon's  Forge  Practice i2mo,  i  50 

Baldwin's  Steam  Heating  for  Buildings i2mo,  2  50 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*  Bartlett's  Mechanical  Drawing 8vo,  3  oo 

*  "  "  "        Abridged  Ed 8vo,  150 

Benjamin's  Wrinkles  and  Recipes i2mo,  2  oo 

Carpenter's  Experimental  Engineering 8vo,  6  oo 

Heating  and  Ventilating  Buildings 8vo,  4  oo 

Cary's  Smoke  Suppression  in  Plants  using  Bituminous  CoaL     (In  Prepara- 
tion.) 

Clerk's  Gas  and  Oil  Engine Small  8vo,  4  oo 

Coolidge's  Manual  of  Drawing 8vo,  paper,  i  oo 

Coolidge  and  Freeman's  Elements  of  General  Drafting  for  Mechanical  En- 
gineers  Oblong  4to,  2  50 

Cromwell's  Treatise  on  Toothed  Gearing ramo,  i  50 

Treatise  on  Belts  and  Pulleys I2mo,  i  50 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Flather's  Dynamometers  and  the  Measurement  of  Power i2mo,  3  oo 

Rope  Driving i2mo,  2  oo 

Gill's  Gas  and  Fuel  Analysis  for  Engineers i2mo,  i  25 

Hall's  Car  Lubrication i2mo,  i  oo 

Bering's  Ready  Reference  Tables  (Conversion  Factors) i6mo,  morocco,  2  50 

12 


Button's  The  Gas  Engine 8vo,  5  oo 

Jamison's  Mechanical  Drawing 8vo,  2  50 

Jones's  Machine  Design: 

Part  I.     Kinematics  of  Machinery. 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kent's  Mechanical  Engineers'  Pocket-book i6mo,  mor«cco,  5  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Leonard's  Machine  Shop,  Tools,  and  Methods 8vo,  4  oo 

*  Lorenz's  Modern  Refrigerating  Machinery.    (Pope,  Haven,  and  Dean.) .  . 8vo,  4  oo 

MacCord's  Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Mechanical  Drawing 4to,  4  oo 

Velocity  Diagrams 8vo,  i  50 

MacFarland's  Standard  Reduction  Factors  for  Gases 8vo,  i  50 

Mahan's  Industrial  Drawing.     (Thompson.) 8vo,  3  50 

Poole's  Calorific  Power  of  Fuels 8vo',  3  oo 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Richard's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  oo 

Smith's  (O.)  Press-working  of  Metals 8vo,  3  oo 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo,  3  oo 

Thurston's   Treatise   on   Friction  and   Lost   Work   in   Machinery   and   Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics.  1 2mo,  i  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's    Kinematics    and    the    Power    of    Transmission.     (Herrmann — 

Klein.) 8vo,  5  oo 

Machinery  of  Transmission  and  Governors.     (Herrmann — Klein.).  .8vo,  5  oo 

Wolff's  Windmill  as  a  Prime  Mover 8vo,  3  oo 

Wood's  Turbines 8vo,  2  50 


MATERIALS   OP   ENGINEERING. 

*  Bovey's  Strength  of  Materials  and  Theory  of  Structures 8vo,  7  50 

Burr's  Elasticity  and  Resistance  of  the  Materials  of  Engineering.    6th  Edition. 

Reset 8vo,  7  50 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Johnson's  Materials  of  Construction 8vo,  6  oo 

Keep's  Cast  Iron 8vo,  2  50 

Lanza's  Applied  Mechanics 8vo,  7  50 

Martens 's  Handbook  on  Testing  Materials.     (Henning.) 8vo,  7  50 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merriman's  Mechanics  of  Materials 8vo,  5  oo 

Strength  of  Materials i2mo,  i  oo 

Metcalf's  Steel.     A  manual  for  Steel-users i2mo,  2  oo 

Sabin's  Industrial  and  Artistic  Technology  of  Paints  and  Varnish 8vo,  3  oo 

Smith's  Materials  of  Machines i2mo,  i  oo 

Thurston's  Materials  of  Engineering 3  vols.,  8vo,  8  oo 

Part  II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Text-book  of  the  Materials  of  Construction 8ve,  5  oo 

Wood's  (De  V.)  Treatise  on  the  Resistance  of  Materials  and  an  Appendix  on 

the  Preservation  of  Timber 8vo,  2  oo 

13 


Wood's  (De  V.)  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Wood's  (M.  P.)  Rustless  Coatings:    Corrosion  and  Electrolysis  of  Iron  and 

SteeL 8vo:  4  oo 

STEAM-ENGINES  AND  BOILERS. 

Berry's  Temperature-entropy  Diagram i2mo,  i  25 

Carnot's  Reflections  on  the  Motive  Power  of  Heat.     (Thurston.) i2mo,  i  50 

Dawson's  "Engineering"  and  Electric  Traction  Pocket-book.  . .  . i6mo,  mor.,  5  oo 

Ford's  Boiler  Making  for  Boiler  Makers i8mo,  i  oo 

Goss's  Locomotive  Sparks 8vo,  2  oo 

Hemenway's  Indicator  Practice  and  Steam-engine  Economy i2mo,  2  oo 

Button's  Mechanical  Engineering  of  Power  Plants 8vo,  5  oo 

Heat  and  Heat-engines .8vo,  5  oo 

Kent's  Steam  boiler  Economy 8vo,  4  oo 

Kneass's  Practice  and  Theory  of  the  Injector 8vo,  i  50 

MacCord's  Slide-valves 8vo,  2  oo 

Meyer's  Modern  Locomotive  Construction •. .  . .  .4to,  10  oc 

Peabody's  Manual  of  the  Steam-engine  Indicator i2mo.  i  50 

Tables  of  the  Properties  of  Saturated  Steam  and  Other  Vapors   8vo,  i  oo 

Thermodynamics  of  the  Steam-engine  and  Other  Heat-engines 8vo,  5  oo 

Valve-gears  for  Steam-engines 8vo,  2  50 

Peabody  and  Miller's  Steam-boilers 8vo,  4  oo 

Pray's  Twenty  Years  with  the  Indicator Large  8vo,  2  50 

Pupin's  Thermodynamics  of  Reversible  Cycles  in  Gases  and  Saturated  Vapors. 

(Osterberg.) i2mo,  i  25 

Reagan's  Locomotives:   Simple   Compound,  and  Electric i2mo,  2  50 

Rontgen's  Principles  of  Thermodynamics.     (Du  Bois.) 8vo,  5  oo 

Sinclair's  Locomotive  Engine  Running  and  Management i2mo,  2  oo 

Smart's  Handbook  of  Engineering  Laboratory  Practice i2mo,  2  50 

Snow's  Steam-boiler  Practice 8vo,  3  oo 

Spangler's  Valve-gears t 8vo,  2  50 

Notes  on  Thermodynamics i2mo,  i  oo 

Spangler,  Greene,  and  Marshall's  Elements  of  Steam-engineering 8vo,  3  oo 

Thomas's  Steam-turbines 8vo,  3  50 

Thurston's  Handy  Tables 8vo,  i  50 

Manual  of  the  Steam-engine 2  vols.,  8vo,  10  oo 

Part  I.     History,  Structure,  and  Theory 8vo,  6  oo 

Part  II.     Design,  Construction,  and  Operation 8vo,  6  oo 

Handbook  of  Engine  and  Boiler  Trials,  and  the  Use  of  the  Indicator  and 

the  Prony  Brake 8vo,  5  oo 

Stationary  Steam-engines 8vo,  2  50 

Steam-boiler  Explosions  in  Theory  and  in  Practice I2mo,  I  50 

Manual  of  Steam-boilers,  their  Designs,  Construction,  and  Operation 8vo,  5  oo 

Weisbach's  Heat,  Steam,  and  Steam-engines.     (Du  Bois.) 8vo,  5  oo 

Whitham's  Steam-engine  Design 8vo,  5  oo 

Wood's  Thermodynamics,  Heat  Motors,  and  Refrigerating  Machines.  .  .8vo,  4  oo 


MECHANICS  AND  MACHINERY. 

Barr's  Kinematics  of  Machinery 8vo,  2  50 

*,Bovey's  Strength  of  Materials  and  Theory  of  Structures   8vo,  7  50 

Chase's  The  Art  of  Pattern-making i2mo,  2  50 

Church's  Mechanics  of  Engineering 8vo,  6  oo 

Notes  and  Examples  in  Mechanics , 8vo,  2  oo 

Compton's  First  Lessons  in  Metal- working i2mo,  i  50 

Compton  and  De  Groodt's  The  Speed  Lathe i2mo  i  50 

14 


Cromwell's  Treatise  on  Toothed  Gearing i2mo,  i  50 

Treatise  on  Belts  and  Pulleys i2mo,  -  50 

Dana's  Text-book  of  Elementary  Mechanics  for  Colleges  and  Schools.  .  i2mo,  i  50 

Dingey's  Machinery  Pattern  Making i2mo,  2  oo 

Dredge's  Record  of  the  Transportation  Exhibits  Building  of  the  World's 

Columbian  Exposition  of  1893 4to  half  morocco,  5  oo 

Du  Bois's  Elementary  Principles  of  Mechanics: 

Vol.      I.     Kinematics 8vo,  3  50 

Vol.    II.     Statics 8vo,  4  oo 

Mechanics  of  Engineering.     Vol.    I. Small  4to,  7  50 

VoL  II Small  4to,  10  oo 

Durley's  Kinematics  of  Machines 8vo,  4  oo 

Fitzgerald's  Boston  Machinist i6mo,  i  oo 

Flather's  Dynamometers,  and  the  Measurement  of  Power i2mo,  3  oo 

Rope  Driving i2mo,  2  oo 

Goss's  Locomotive  Sparks 8vo,  2  oo 

*  Greene's  Structural  Mechanics 8vo,  2  50 

Hall's  Car  Lubrication i2mo,  i  oo 

Holly's  Art  of  Saw  Filing i8mo,  75 

James's  Kinemr.tics  of  a  Point  and  the  Rational  Mechanics  of  a  Particle. 

Small  8vo,  2  oo 

*  Johnson's  (W.  W.)  Theoretical  Mechanics. . i2mo,  3  oo 

Johnson's  (L.  J.)  Statics  by  Graphic  and  Algebraic  Methods 8vo,  2  oo 

Jones's  Machine  Design: 

Part    I.     Kinematics  of  Machinery 8vo,  i  50 

Part  II.     Form,  Strength,  and  Proportions  of  Parts 8vo,  3  oo 

Kerr's  Power  and  Power  Transmission 8vo,  2  oo 

Lanza's  Applied  Mechanics 8vo,  7  50 

Leonard's  Machine  Shop,  Tools,  and  Methods 8vo,  4  oo 

*  Lorenz's  Modern  Refrigerating  Machinery.     (Pope,  Haven,  and  Dean.).8vo,  4  oo 
MacCord's  Kinematics;  or,  Practical  Mechanism 8vo,  5  oo 

Velocity  Diagrams 8vo,  i  50 

Maurer's  Technical  Mechanics 8vo,  4  oo 

Merriman's  Mechanics  of  Materials . 8vo,  5  oo 

*  Elements  of  Mechanics i2mo,  i  oo 

*  Michie's  Elements  of  Analytical  Mechanics 8vo,  4  oc 

Reagan's  Locomotives:   Simple,  Compound,  and  Electric i2mo,  2  50 

Reid's  Course  in  Mechanical  Drawing 8vo,  2  oo 

Text-book  of  Mechanical  Drawing  and  Elementary  Machine  Design. 8vo,  3  oo 

Richards's  Compressed  Air i2mo,  i  50 

Robinson's  Principles  of  Mechanism 8vo,  3  oo 

Ryan,  Norris,  and  Hoxie's  Electrical  Machinery.     Vol.  1 8vo,  2  50 

Schwamb  and  Merrill's  Elements  of  Mechanism 8vo,  3  co 

Sinclair's  Locomotive-engine  Running  and  Management I2mo,  2  oo 

Smith's  (O.)  Press-working  of  Metals 8vo,  3  oo 

Smith's  (A.  W.)  Materials  of  Machines i2mo,  i  oo 

Smith  (A.  W.)  and  Marx's  Machine  Design 8vo,  3  oo 

Spangler,  Greent.and  Marshall's  Elements  of  Steam-engineering 8vo,  3  oo 

Thurston's  Treatise  on  Friction  and  Lost  Work  in    Machinery  and    Mill 

Work 8vo,  3  oo 

Animal  as  a  Machine  and  Prime  Motor,  and  the  Laws  of  Energetics. 

i2mo,  i  oo 

Warren's  Elements  of  Machine  Construction  and  Drawing 8vo,  7  50 

Weisbach's  Kinematics  and  Power  of  Transmission.   (Herrmann — Klein.  ).8vo,  5  oo 

Machinery  of  Transmission  and  Governors.      (Herrmann — Klein.). 8vo,  5  oo 

Wood's  Elements  of  Analytical  Mechanics 8vo,  3  oo 

Principles  of  Elementary  Mechanics i2mo,  i  25 

Turbines. 8vo,  2  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  oo 

15 


METALLURGY. 


Egleston's  Metallurgy  of  Silver,  Gold,  and  Mercury: 

Vol.    I.     Silver 8vo,  7  50 

.     Vol.  II.     Gold  and  Mercury 8vo,  7  50 

**  Iles's  Lead-smelting.     (Postage  9  cents  additional.) I2mo,  2  50 

Keep's  Cast  Iron 8vo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

Le  Chatelier's  High-temperature  Measurements.  (Boudouard — Burgess. )i2nao.  3  oo 

Metcalf's  Steel.     A  Manual  for  Steel-users i2mo,  2  oo 

Minet's  Production  of  Aluminum  and  its  Industrial  Use.     (Waldo.). . .  .  i2mo,  2  50 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo,  4  oo 

Smith's  Materials  of  Machines I2mo,  i  oo 

Thurston's  Materials  of  Engineering.     In  Three  Parts. 8vo,  8  oo 

Part    II.     Iron  and  Steel 8vo,  3  50 

Part  III.     A  Treatise  on  Brasses,  Bronzes,  and  Other  Alloys  and  their 

Constituents 8vo,  2  50 

Ulke's  Modern  Electrolytic  Copper  Refining 8vo,  3  oo 


MINERALOGY. 

Barringer's  Description  of  Minerals  of  Commercial  Value.    Oblong,  morocco,  2  50 

Boyd's  Resources  of  Southwest  Virginia 8vo,  3  oo 

Map  of  Southwest  Virignia Pocket-book  form.  2  oo 

Brush's  Manual  of  Determinative  Mineralogy.     (Penfield.) 8vo,  4  oo 

Chester's  Catalogue  of  Minerals 8vo,  paper,  i  oo 

Cloth,  i  25 

Dictionary  of  the  Names  of  Minerals 8vo,  3  50 

Dana's  System  of  Mineralogy Large  8vo,  half  leather,  12  50 

First  Appendix  to  Dana's  New  "  System  of  Mineralogy." Large  8vo,  i  oo 

Text-book  of  Mineralogy 8vo,  4  oo 

Minerals  and  How  to  Study  Them 121110.  i  50 

Catalogue  of  American  Localities  of  Minerals Large  8vo,  i  oo 

Manual  of  Mineralogy  and  Petrography I2mo,  2  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects I2mo,  i  oo 

Eakle's  Mineral  Tables 8vo,  i  25 

Egleston's  Catalogue  of  Minerals  and  Synonyms 8vo,  2  50 

Hussak's  The  Determination  of  Rock-forming  Minerals.    (Smith.). Small 8vo,  2  oo 

Merrill's  Non-metallic  Minerals:   Their  Occurrence  and  Uses 8vo,  4  oo 

*  Penfield's  Notes  on  Determinative  Mineralogy  and  Record  of  Mineral  Tests. 

8vo,  paper,  50 
Rosenbusch's   Microscopical  Physiography   of   the   Rock-making  Minerals. 

(Iddings.) 8vo,  5  oo 

*  Tillman's  Text-book  of  Important  Minerals  and  Rocks 8vo,  2  oo 


MINING. 

Beard's  Ventilation  of  Mines i2mo,  2  50 

Boyd's  Resources  of  Southwest  Virginia Fvo,  3  oo 

Map  of  Southwest  Virginia Pocket-book  form  2  oo 

Douglas's  Untechnical  Addresses  on  Technical  Subjects i2mo  i  oo 

*  Drinker's  Tunneling,  Explosive  Compounds,  and  Roc'.s  Drills.   4to,hf.  mor.,  25  oo 

Eissler's  Modern  High  Explosives 8vo,  4  oo 

16 


Goodyear's  Coal-mines  of  the  Western  Coast  of  the  United  States I2mo,  2  50 

Ihlseng's  Manual  of  Mining 8vo,  5  oo 

**  Iles's  Lead-smelting.     (Postage  QC.  additional.) i2mo,  2  50 

Kunhardt's  Practice  of  Ore  Dressing  in  Europe 8vo,  i  50 

O'Driscoll's  Notes  on  the  Treatment  of  Gold  Ores 8vo,  2  oo 

Robine  and  Lenglen's  Cyanide  Industry.     (Le  Clerc.) 8vo,  4  oo 

*  Walke's  Lectures  on  Explosives 8vo,  4  oo 

Wilson's  Cyanide  Processes 12 mo,  i  50 

Chlorination  Process I2mo,  i  50 

Hydraulic  and  Placer  Mining I2mo,  2  oo 

Treatise  on  Practical  and  Theoretical  Mine  Ventilation i2mo,  i  25 


SANITARY  SCIENCE. 

Bashore's  Sanitation  0f  a  Country  House I2mo,  i  oo 

Folwell's  Sewerage.     (Designing,  Construction,  and  Maintenance.) 8vo,  3  oo 

Water-supply  Engineering 8vo,  4  oo 

Fowler's  Sewage  Works  Analyses I2mo,  2  oo 

Fuertes's  Water  and  Public  Health I2mo,  i  50 

Water-filtration  Works I2mo,  2  50 

Gerhard's  Guide  to  Sanitary  House-inspection i6mo,  i  oo 

Goodrich's  Economic  Disposal  of  Town's  Refuse Demy  8vo,  3  50 

Hazen's  Filtration  of  Public  Water-supplies 8vo,  3  oo 

Leach's  The  Inspection  and  Analysis  of  Food  with  Special  Reference  to  State 

Control 8vo,  7  50 

Mason's  Water-supply.  (Considered  principally  from  a  Sanitary  Standpoint)  8vo,  4  oo 

Examination  of  Water.     (Chemical  and  Bacteriological.) i2mo,  i  25 

Ogden's  Sewer  Design i2mo,  2  oo 

Prescott  and  Winslow's  Elements  of  Water  Bacteriology,  with  Special  Refer- 
ence to  Sanitary  Water  Analysis i2mo,  i  25 

*  Price's  Handbook  on  Sanitation i2mo,  i  50 

Richards's  Cost  of  Food.     A  Study  in  Dietaries i2mo,  i  oo 

Cost  of  Living  as  Modified  by  Sanitary  Science i2mo,  i  oo 

Cost  of  Shelter i2mo,  i  oo 

Richards  and  Woodman's  Air,  Water,  and  Food  from  a  Sanitary  Stand- 
point  8vo,  2  oo 

*  Richards  and  Wiliiams's  The  Dietary  Computer 8vo,  i  50 

Rideal's  Sewage  and  Bacterial  Purification  of  Sewage 8vo,  3  50 

Turneaure  and  Russell's  Public  Water-supplies 8vo,  5  oo 

Von  Behring's  Suppression  of  Tuberculosis.     (Bclduan.) i2mo,  i  oo 

Whipple's  Microscopy  of  Drinking-water 8vo,  3  50 

Winton's  Microscopy  of  Vegetable  Foods 8vo,  7  50 

Woodhull's  Notes  on  Military  Hygiene i6mo,  i  50 

*  Personal  H/giene I2mo,  i  oo 


MISCELLANEOUS. 

De  Fursac's  Manual  of  Psychiatry.     (Rosanoff  and  Collins.).  .  .  .Large  i2mo,  2  50 
Emmons's  Geological  Guide-took  of  the  Rocky  Mountain  Excursion  of  the 

International  Congress  of  Geologists Large  £vo,  i  50 

Ferrel's  Popular  Treatise  on  the  Winds 8vo  4  oo 

Raines's  American  Railway  Management i2mo,  2  50 

Mott's  Fallacy  of  the  Present  Theory  of  Sound i6mo,  i  oo 

Ricketts's  History  of  Rensselaer  Polytechnic  Institute,  1824-1894. .Small  8vo,  3  oc 

Rostoski's  Serum  Diagnosis.     (Bolduan.) i2mo.  i  oo 

Rotherham's  Emphasized  New  Testament Large  8vo,  2  oo 

17 


Steel's  Treatise  on  the  Diseases  of  the  Dog 8vo,  3  50 

The  World's  Columbian  Exposition  of  1893 4to,  i  oo 

Von  Behring's  Suppression  of  Tuberculosis.     (Bolduan.) i2mo,  i  oo 

Winslow's  Elements  of  Applied  Microscopy iimo,  I  50 

Worcester  and  Atkinson.     Small  Hospitals,  Establishment  and  Maintenance; 

Suggestions  for  Hospital  Architecture :  Plans  for  Small  Hospital .  1 2 mo ,  i  25 


HEBREW  AND  CHALDEE  TEXT-BOOKS. 


Green's  Elementary  Hebrew  Grammar i2mo,  i  25 

Hebrew  Chrestomathy 8vo,  2  oo 

Gesenius's  Hebrew  and  Chaldee  Lexicon  to  the  Old  Testament  Scriptures. 

(Tregelles.) Small  4to,  half  morocco,  5  oo 

Letteris's  Hebrew  Bible. 8vo,  2  25 

18 


t 


UNIVERSITY  OP   CALIFORNIA  LIBRARY 

RETURN  TO  the  circulation  desk  of  any 
University  of  California  Library 

or  to  the 

NORTHERN  REGIONAL  LIBRARY  FACILITY 
Bldg.  400,  Richmond  Field  Station 
University  of  California 
Richmond,  CA  94804-4698 

ALL  BOOKS  MAY  BE  RECALLED  AFTER  7  DAYS 

•  2-month  loans  may  be  renewed  by  calling 
(510)642-6753 

•  1-year  loans  may  be  recharged  by  bringing 
books  to  NRLF 

•  Renewals  and  recharges  may  be  made  4 
days  prior  to  due  date. 

DUE  AS  STAMPED  BELOW 


MQ\f  3 


12,000(11/95) 


